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@@ -39,8 +39,8 @@ % Our mysterious boy \def\urusi{URu$_{\text2}$Si$_{\text2}$} -\def\e{{\text{\textsc{elastic}}}} % "elastic" -\def\i{{\text{\textsc{int}}}} % "interaction" +\def\ee{{\text{\textsc{elastic}}}} % "elastic" +\def\ii{{\text{\textsc{int}}}} % "interaction" \def\Dfh{D$_{\text{4h}}$} @@ -114,25 +114,25 @@ superconductivity and \ho\ give way to local moment antiferromagnetism Kusunose_2011_On, Kung_2015, Cricchio_2009, Ohkawa_1999, Santini_1994, Kiss_2005, Harima_2010, Thalmeier_2011, Tonegawa_2012_Cyclotron, Rau_2012_Hidden, Riggs_2015_Evidence, Hoshino_2013_Resolution, -Ikeda_1998_Theory, Chandra_2013_Hastatic, 1902.06588v2, Ikeda_2012} propose +Ikeda_1998_Theory, Chandra_2013_Hastatic, Harrison_2019_Hidden, Ikeda_2012} propose associating any of a variety of broken symmetries with \ho. This work analyzes a family of phenomenological models with order parameters of general symmetry that couple linearly to strain. Of these, only one is compatible with two experimental observations: first, the $\Bog$ ``nematic" elastic susceptibility $(C_{11}-C_{12})/2$ softens anomalously from room temperature down to -$T_{\text{\ho}}=17.5\,\K$;\cite{deVisser_1986} and second, a $\Bog$ nematic +$T_{\text{\ho}}=17.5\,\K$;\cite{deVisser_1986_Thermal} and second, a $\Bog$ nematic distortion is observed by x-ray scattering under sufficient pressure to destroy the \ho\ state.\cite{Choi_2018} Recent resonant ultrasound spectroscopy (\rus) measurements were used to examine the thermodynamic discontinuities in the elastic moduli at -$T_{\text{\ho}}$.\cite{1903.00552v1} The observation of discontinues only in -compressional, or $\Aog$, elastic moduli requires that the point-group -representation of \ho\ be one-dimensional. This rules out many order parameter -candidates~\cite{Thalmeier_2011, Tonegawa_2012_Cyclotron, Rau_2012_Hidden, -Riggs_2015_Evidence, Hoshino_2013_Resolution, Ikeda_2012, Chandra_2013_Origin} -in a model-independent way, but doesn't differentiate between those that -remain. +$T_{\text{\ho}}$.\cite{Ghosh_2020_One-component} The observation of +discontinuities only in compressional, or $\Aog$, elastic moduli requires that +the point-group representation of \ho\ be one-dimensional. This rules out many +order parameter candidates~\cite{Thalmeier_2011, Tonegawa_2012_Cyclotron, +Rau_2012_Hidden, Riggs_2015_Evidence, Hoshino_2013_Resolution, Ikeda_2012, +Chandra_2013_Origin} in a model-independent way, but doesn't differentiate +between those that remain. Recent x-ray experiments discovered rotational symmetry breaking in \urusi\ under pressure.\cite{Choi_2018} Above 0.13--0.5 $\GPa$ (depending on @@ -164,11 +164,11 @@ The point group of \urusi\ is \Dfh, and any 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 $f_\e=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$. +the strain $\epsilon$ is $f_\ee=C^0_{ijkl}\epsilon_{ij}\epsilon_{kl}$. \footnote{Components of the elastic modulus tensor $C$ were given in the popular Voigt notation in the abstract and introduction. Here and henceforth the notation used is that natural for a rank-four tensor.} The form of the bare -moduli tensor $C^0$ is further restricted by symmetry. Linear combinations of +moduli tensor $C^0$ is further restricted by symmetry. \cite{Landau_1986_Theory} Linear combinations of the six independent components of strain form five irreducible components of strain in \Dfh\ as \begin{equation} @@ -184,7 +184,7 @@ strain in \Dfh\ as 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^0_{\X,ij}\epsilon_{\X,i}\epsilon_{\X,j}, + f_\ee=\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 $C^0_\X$ are @@ -202,7 +202,7 @@ 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. + f_\ii=-b^{(i)}\epsilon_\X^{(i)}\eta. \end{equation} Many high-order interations are permitted, and in the appendix another of the form $\epsilon^2\eta^2$ is added to the following analysis. @@ -211,7 +211,7 @@ $\X$ then there can be no linear coupling. The next-order coupling is linear in strain, quadratic in order parameter, and the effect of this coupling at a continuous phase transition is to produce a jump in the $\Aog$ elastic moduli if $\eta$ is single-component, \cite{Luthi_1970, Ramshaw_2015, Shekhter_2013} -and jumps in other elastic moduli if multicomponent.\cite{1903.00552v1} Because +and jumps in other elastic moduli if multicomponent.\cite{Ghosh_2020_One-component} Because we are interested in physics that anticipates the phase transition---for instance, that the growing \op\ susceptibility is reflected directly in the elastic susceptibility---we will focus our attention on \op s that can produce @@ -223,7 +223,7 @@ representations. If the \op\ transforms like $\Aog$ (e.g. a fluctuation in valence number), odd terms are allowed in its free energy and without fine-tuning any transition will be first order and not continuous. Since the \ho\ phase transition is -second-order,\cite{deVisser_1986} we will henceforth rule out $\Aog$ \op s as +second-order,\cite{deVisser_1986_Thermal} we will henceforth rule out $\Aog$ \op s as well. For the \op\ representation $\X$ as any of those remaining---$\Bog$, $\Btg$, or $\Eg$---the most general quadratic free energy density is \begin{equation} @@ -242,8 +242,8 @@ full free energy functional of $\eta$ and $\epsilon$ is \begin{equation} \begin{aligned} F[\eta,\epsilon] - &=F_\op[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\ - &=\int dx\,(f_\op+f_\e+f_\i). + &=F_\op[\eta]+F_\ee[\epsilon]+F_\ii[\eta,\epsilon] \\ + &=\int dx\,(f_\op+f_\ee+f_\ii). \end{aligned} \label{eq:free_energy} \end{equation} @@ -473,7 +473,7 @@ corresponding modulus. \includegraphics{fig-stiffnesses} \caption{ \Rus\ measurements of the elastic moduli of \urusi\ at ambient pressure as a - function of temperature from recent experiments\cite{1903.00552v1} (blue, + function of temperature from recent experiments\cite{Ghosh_2020_One-component} (blue, solid) alongside fits to theory (magenta, dashed and black, solid). The solid yellow region shows the location of the \ho\ phase. (a) $\Btg$ modulus data and a fit to the standard form.\cite{Varshni_1970} (b) $\Bog$ modulus data and a fit to @@ -491,7 +491,7 @@ corresponding modulus. \label{fig:data} \end{figure*} -\Rus\ experiments~\cite{1903.00552v1} yield the individual elastic moduli +\Rus\ experiments~\cite{Ghosh_2020_One-component} yield the individual elastic moduli broken into irreps; data for the $\Bog$ and $\Btg$ components defined in \eqref{eq:strain-components} are shown in Figures \ref{fig:data}(a--b). The $\Btg$ in Fig.~\ref{fig:data}(a) modulus doesn't appear to have any response to @@ -540,7 +540,7 @@ from the Lifshitz point we expect the wavevector to lock into values commensurate with the space group of the lattice, and moreover that at zero pressure, where the \rus\ data here was collected, the half-wavelength of the modulation should be commensurate with the lattice spacing $a_3\simeq9.68\,\A$, -or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{Meng_2013, Broholm_1991, Wiebe_2007, +or $q_*=\pi/a_3\simeq0.328\,\A^{-1}$.\cite{Meng_2013_Imaging, Broholm_1991, Wiebe_2007, Bourdarot_2010, Hassinger_2010} In between these two regimes, mean field theory predicts that the ordering wavevector shrinks by jumping between ever-closer commensurate values in the style of the devil's staircase.\cite{Bak_1982} In @@ -552,7 +552,7 @@ commensurability jumps) at low pressure and approach zero as $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. Alternatively, \rus\ done at ambient pressure might examine the heavy Fermi liquid to \afm\ transition by doping. Though previous \rus\ studies have doped \urusi\ with -Rhodium,\cite{Yanagisawa_2014} the magnetic rhodium dopants likely promote +rhodium,\cite{Yanagisawa_2014} the magnetic rhodium dopants likely promote magnetic phases. A non-magnetic dopant such as phosphorous may more faithfully explore the transition out of the HO phase. Our work also motivates experiments that can probe the entire correlation function---like x-ray and neutron @@ -585,18 +585,16 @@ at the Ginzburg temperature. \cite{Hornreich_1980, Ginzburg_1961_Some} Magnetic phase transitions tend to have a Ginzburg temperature of order one. Our fit above gives $\xi_{\perp0}q_*=(D_\perp q_*^4/aT_c)^{1/4}\simeq2$, which combined with the speculation of $q_*\simeq\pi/a_3$ puts the bare correlation length -$\xi_{\perp0}$ at about what one would expect for a generic magnetic -transition. The agreement of this data in the $t\sim0.1$--10 range with the -mean field exponent suggests that this region is outside the Ginzburg region, -but an experiment may begin to see deviations from mean field behavior within -approximately several Kelvin of the critical point. An ultrasound experiment -with more precise temperature resolution near the critical point may be able to -resolve a modified cusp exponent $\gamma\simeq1.31$,\cite{Guida_1998_Critical} since the -universality class of a uniaxial modulated one-component \op\ is $\mathrm -O(2)$.\cite{Garel_1976} We should not expect any quantitative 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. +$\xi_{\perp0}$ on the order of lattice constant, which is about what one would +expect for a generic magnetic transition. The agreement of this data in the +$t\sim0.1$--10 range with the mean field exponent suggests that this region is +outside the Ginzburg region, but an experiment may begin to see deviations from +mean field behavior within approximately several Kelvin of the critical point. +An ultrasound experiment with more precise temperature resolution near the +critical point may be able to resolve a modified cusp exponent +$\gamma\simeq1.31$,\cite{Guida_1998_Critical} since according to one analysis +the universality class of a uniaxial modulated one-component \op\ is that of +the $\mathrm O(2)$, 3D XY transition.\cite{Garel_1976} \section{Conclusion and Outlook.} We have developed a general phenomenological treatment of \ho\ \op s that have the potential for linear coupling to strain. @@ -609,32 +607,14 @@ 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 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}. -} +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 \afm\ is compelling, but our mean field theory does not make any explicit @@ -669,11 +649,11 @@ such as ultrasound, that could further support or falsify this idea. In this appendix, we compute the $\Bog$ modulus for a theory with a high-order interaction truncation to better match the low-temperature behavior. Consider -the free energy density $f=f_\e+f_\i+f_\op$ with +the free energy density $f=f_\ee+f_\ii+f_\op$ with \begin{equation} \begin{aligned} - f_\e&=\frac12C_0\epsilon^2 \\ - f_\i&=-b\epsilon\eta+\frac12g\epsilon^2\eta^2 \\ + f_\ee&=\frac12C_0\epsilon^2 \\ + f_\ii&=-b\epsilon\eta+\frac12g\epsilon^2\eta^2 \\ f_\op&=\frac12\big[r\eta^2+c_\parallel(\nabla_\parallel\eta)^2+c_\perp(\nabla_\perp\eta)^2+D(\nabla_\perp^2\eta)^2\big]+u\eta^4. \end{aligned} \label{eq:new_free_energy} @@ -711,8 +691,8 @@ where \begin{equation} C(x,x') =\frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}\bigg|_{\epsilon=\langle\epsilon\rangle} - =\frac{\delta^2F_\e[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}+ - \frac{\delta^2F_\i[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}+ + =\frac{\delta^2F_\ee[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}+ + \frac{\delta^2F_\ii[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')}+ \frac{\delta^2F_\op[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')} \bigg|_{\epsilon=\langle\epsilon\rangle} \end{equation} @@ -724,14 +704,14 @@ and $\eta_\star$ is the mean-field order parameter conditioned on the strain def \end{equation} We will work this out term by term. The elastic term is the most straightforward, giving \begin{equation} - \frac{\delta^2F_\e[\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')} + \frac{\delta^2F_\ee[\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')} =\frac12C_0\frac{\delta^2}{\delta\epsilon(x)\delta\epsilon(x')}\int dx''\,\epsilon(x'')^2 =C_0\delta(x-x'). \end{equation} The interaction term gives \begin{equation} \begin{aligned} - \frac{\delta^2F_\i[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')} + \frac{\delta^2F_\ii[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon(x)\delta\epsilon(x')} &=-b\frac{\delta^2}{\delta\epsilon(x)\delta\epsilon(x')}\int dx''\,\epsilon(x'')\eta_\star[\epsilon](x'') +\frac12g\frac{\delta^2}{\delta\epsilon(x)\delta\epsilon(x')}\int dx''\,\epsilon(x'')^2\eta_\star[\epsilon](x'')^2 \\ &=-b\frac{\delta\eta_\star[\epsilon](x')}{\delta\epsilon(x)} |