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diff --git a/hidden_order.bib b/hidden_order.bib index 2194a59..501b56c 100644 --- a/hidden_order.bib +++ b/hidden_order.bib @@ -1,47 +1,4 @@ -@book{landau_theory_1995, - series = {Landau and {{Lifshitz Course}} of {{Theoretical Physics}}}, - title = {Theory of {{Elasticity}}}, - author = {Landau, Lev Davidovich and Lifshitz, Eugin M and Berestetskii, VB and Pitaevskii, LP}, - year = {1995}, - keywords = {_tablet}, - file = {/home/pants/Zotero/storage/AQ7G8AHB/Landau et al. - 1995 - Theory of Elasticity.pdf} -} - -@article{ginzburg_remarks_1961, - title = {Some {{Remarks}} on {{Phase Transitions}} of the {{Second Kind}} and the {{Microscopic}} Theory of {{Ferroelectric Materials}}}, - volume = {2}, - number = {9}, - journal = {Soviet Physics, Solid State}, - author = {Ginzburg, V. L.}, - year = {1961}, - keywords = {⛔ No DOI found}, - pages = {1824-1834}, - file = {/home/pants/Zotero/storage/JVMTIZGB/Ginzburg - 1961 - Some Remarks on Phase Transitions of the Second Ki.pdf} -} - -@article{lifshitz_theory_1942, - title = {On the Theory of Phase Transitions of the Second Order {{I}}. {{Changes}} of the Elementary Cell of a Crystal in Phase Transitions of the Second Order}, - volume = {6}, - journal = {Proceedings of the USSR Academy of Sciences Journal of Physics}, - author = {Lifshitz, EM}, - year = {1942}, - keywords = {_tablet,⛔ No DOI found}, - pages = {61}, - file = {/home/pants/Zotero/storage/D9BYG3FK/Lifshitz - 1942 - On the theory of phase transitions of the second o.pdf} -} - -@article{lifshitz_theory_1942-1, - title = {On the Theory of Phase Transitions of the Second Order {{II}}. {{Phase}} Transitions of the Second Order in Alloys}, - volume = {6}, - journal = {Proceedings of the USSR Academy of Sciences Journal of Physics}, - author = {Lifshitz, EM}, - year = {1942}, - keywords = {_tablet,⛔ No DOI found}, - pages = {251}, - file = {/home/pants/Zotero/storage/TAA9G46H/Lifshitz - 1942 - On the theory of phase transitions of the second o.pdf} -} - @article{el-showk_solving_2014, title = {Solving the 3d {{Ising Model}} with the {{Conformal Bootstrap II}}. {$\mathsl{c}$}-{{Minimization}} and {{Preise Critial Exponents}}}, volume = {157}, @@ -56,7 +13,32 @@ year = {2014}, keywords = {_tablet}, pages = {869-914}, - file = {/home/pants/Zotero/storage/XB5EWQ28/El-Showk et al. - 2014 - Solving the 3d Ising Model with the Conformal Boot.pdf} + file = {/home/pants/.zotero/data/storage/XB5EWQ28/El-Showk et al. - 2014 - Solving the 3d Ising Model with the Conformal Boot.pdf} +} + +@article{guida_critical_1998, + title = {Critical Exponents of the {{N}}-Vector Model}, + volume = {31}, + issn = {0305-4470}, + abstract = {Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img1.gif] and \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img2.gif] ) of the three-dimensional \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img3.gif] field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N -vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img4.gif] (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined. The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values. Finally, because an error has been discovered in the last order of the published \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img5.gif] expansions (order \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img6.gif] ), we have also re-analysed the determination of exponents from the \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion. The conclusion is that the general agreement between \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.}, + language = {en}, + number = {40}, + journal = {Journal of Physics A: Mathematical and General}, + doi = {10.1088/0305-4470/31/40/006}, + author = {Guida, R. and {Zinn-Justin}, J.}, + year = {1998}, + keywords = {_tablet}, + pages = {8103}, + file = {/home/pants/.zotero/data/storage/K468APXL/Guida and Zinn-Justin - 1998 - Critical exponents of the N-vector model.pdf} +} + +@book{landau_theory_1995, + series = {Landau and {{Lifshitz Course}} of {{Theoretical Physics}}}, + title = {Theory of {{Elasticity}}}, + author = {Landau, Lev Davidovich and Lifshitz, Eugin M and Berestetskii, VB and Pitaevskii, LP}, + year = {1995}, + keywords = {_tablet}, + file = {/home/pants/.zotero/data/storage/AQ7G8AHB/Landau et al. - 1995 - Theory of Elasticity.pdf} } @article{fisher_specific_1990, @@ -73,7 +55,7 @@ year = {1990}, keywords = {_tablet}, pages = {419-423}, - file = {/home/pants/Zotero/storage/HHVDKMSP/Fisher et al. - 1990 - Specific heat of URu₂Si₂ Effect of pressure and m.pdf} + file = {/home/pants/.zotero/data/storage/HHVDKMSP/Fisher et al. - 1990 - Specific heat of URu₂Si₂ Effect of pressure and m.pdf} } @article{hornreich_lifshitz_1980, @@ -89,67 +71,70 @@ year = {1980}, keywords = {_tablet}, pages = {387-392}, - file = {/home/pants/Zotero/storage/FQWHY9TF/Hornreich - 1980 - The Lifshitz point Phase diagrams and critical be.pdf} + file = {/home/pants/.zotero/data/storage/FQWHY9TF/Hornreich - 1980 - The Lifshitz point Phase diagrams and critical be.pdf} } -@article{guida_critical_1998, - title = {Critical Exponents of the {{N}}-Vector Model}, - volume = {31}, - issn = {0305-4470}, - abstract = {Recently the series for two renormalization group functions (corresponding to the anomalous dimensions of the fields \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img1.gif] and \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img2.gif] ) of the three-dimensional \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img3.gif] field theory have been extended to next order (seven loops) by Murray and Nickel. We examine the influence of these additional terms on the estimates of critical exponents of the N -vector model, using some new ideas in the context of the Borel summation techniques. The estimates have slightly changed, but remain within the errors of the previous evaluation. Exponents such as \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img4.gif] (related to the field anomalous dimension), which were poorly determined in the previous evaluation of Le Guillou-Zinn-Justin, have seen their apparent errors significantly decrease. More importantly, perhaps, summation errors are better determined. The change in exponents affects the recently determined ratios of amplitudes and we report the corresponding new values. Finally, because an error has been discovered in the last order of the published \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img5.gif] expansions (order \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img6.gif] ), we have also re-analysed the determination of exponents from the \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion. The conclusion is that the general agreement between \#\#IMG\#\# [http://ej.iop.org/images/0305-4470/31/40/006/img7.gif] -expansion and three-dimensional series has improved with respect to Le Guillou-Zinn-Justin.}, - language = {en}, - number = {40}, - journal = {Journal of Physics A: Mathematical and General}, - doi = {10.1088/0305-4470/31/40/006}, - author = {Guida, R. and {Zinn-Justin}, J.}, - year = {1998}, - keywords = {_tablet}, - pages = {8103}, - file = {/home/pants/Zotero/storage/K468APXL/Guida and Zinn-Justin - 1998 - Critical exponents of the N-vector model.pdf} +@article{lifshitz_theory_1942-1, + title = {On the Theory of Phase Transitions of the Second Order {{II}}. {{Phase}} Transitions of the Second Order in Alloys}, + volume = {6}, + journal = {Proceedings of the USSR Academy of Sciences Journal of Physics}, + author = {Lifshitz, EM}, + year = {1942}, + keywords = {⛔ No DOI found,_tablet}, + pages = {251}, + file = {/home/pants/.zotero/data/storage/TAA9G46H/Lifshitz - 1942 - On the theory of phase transitions of the second o.pdf} } -@article{varshni_temperature_1970, - title = {Temperature {{Dependence}} of the {{Elastic Constants}}}, +@article{lifshitz_theory_1942, + title = {On the Theory of Phase Transitions of the Second Order {{I}}. {{Changes}} of the Elementary Cell of a Crystal in Phase Transitions of the Second Order}, + volume = {6}, + journal = {Proceedings of the USSR Academy of Sciences Journal of Physics}, + author = {Lifshitz, EM}, + year = {1942}, + keywords = {⛔ No DOI found,_tablet}, + pages = {61}, + file = {/home/pants/.zotero/data/storage/D9BYG3FK/Lifshitz - 1942 - On the theory of phase transitions of the second o.pdf} +} + +@article{ginzburg_remarks_1961, + title = {Some {{Remarks}} on {{Phase Transitions}} of the {{Second Kind}} and the {{Microscopic}} Theory of {{Ferroelectric Materials}}}, volume = {2}, - abstract = {The following two equations are proposed for the temperature dependence of the elastic stiffness constants: cij=c0ij-s(etT-1) and cij=a-bT2(T+c), where c0ij, s, t, a, b, and c are constants. The applicability of these two equations and that of Wachtman's equation is examined for 57 elastic constants of 22 substances. The first equation has a theoretical justification and gives the best over-all results. Neither of the three equations give the theoretically expected T4 dependence at low temperatures, and therefore they are not expected to give very accurate results at very low temperatures ({$\lessequivlnt\Theta$}D50). A new melting criterion is also examined.}, - number = {10}, - journal = {Physical Review B}, - doi = {10.1103/physrevb.2.3952}, - author = {Varshni, Y. P.}, - month = nov, - year = {1970}, - pages = {3952-3958}, - file = {/home/pants/Zotero/storage/QN7TLJV7/Varshni - 1970 - Temperature Dependence of the Elastic Constants.pdf} + number = {9}, + journal = {Soviet Physics, Solid State}, + author = {Ginzburg, V. L.}, + year = {1961}, + keywords = {⛔ No DOI found}, + pages = {1824-1834}, + file = {/home/pants/.zotero/data/storage/JVMTIZGB/Ginzburg - 1961 - Some Remarks on Phase Transitions of the Second Ki.pdf} } -@article{kiss_group_2005, - title = {Group Theory and Octupolar Order in \$\textbackslash{}mathrm\{\vphantom\}{{U}}\vphantom\{\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, - volume = {71}, - abstract = {Recent experiments on URu2Si2URu2Si2 show that the low-pressure hidden order is nonmagnetic but it breaks time reversal invariance. Restricting our attention to local order parameters of 5f25f2 shells, we find that the best candidate for hidden order is staggered order of either Tz{$\beta$}T{$\beta$}z or TxyzTxyz octupoles. Group theoretical arguments for the effect of symmetry-lowering perturbations (magnetic field, mechanical stress) predict behavior in good overall agreement with observations. We illustrate our general arguments on the example of a five-state crystal field model which differs in several details from models discussed in the literature. The general appearance of the mean field phase diagram agrees with the experimental results. In particular, we find that (a) at zero magnetic field, there is a first-order phase boundary between octupolar order and large-moment antiferromagnetism with increasing hydrostatic pressure; (b) arbitrarily weak uniaxial pressure induces staggered magnetic moments in the octupolar phase; and (c) a new phase with different symmetry appears at large magnetic fields.}, - number = {5}, +@article{choi_pressure-induced_2018, + title = {Pressure-Induced Rotational Symmetry Breaking in \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, + volume = {98}, + abstract = {Phase transitions and symmetry are intimately linked. Melting of ice, for example, restores translation invariance. The mysterious hidden order (HO) phase of URu2Si2 has, despite relentless research efforts, kept its symmetry breaking element intangible. Here, we present a high-resolution x-ray diffraction study of the URu2Si2 crystal structure as a function of hydrostatic pressure. Below a critical pressure threshold pc{$\approx$}3 kbar, no tetragonal lattice symmetry breaking is observed even below the HO transition THO=17.5 K. For p{$>$}pc, however, a pressure-induced rotational symmetry breaking is identified with an onset temperatures TOR{$\sim$}100 K. The emergence of an orthorhombic phase is found and discussed in terms of an electronic nematic order that appears unrelated to the HO, but with possible relevance for the pressure-induced antiferromagnetic (AF) phase. Existing theories describe the HO and AF phases through an adiabatic continuity of a complex order parameter. Since none of these theories predicts a pressure-induced nematic order, our finding adds an additional symmetry breaking element to this long-standing problem.}, + number = {24}, journal = {Physical Review B}, - doi = {10.1103/physrevb.71.054415}, - author = {Kiss, Annam{\'a}ria and Fazekas, Patrik}, - month = feb, - year = {2005}, - pages = {054415}, - file = {/home/pants/Zotero/storage/YTARVDIM/Kiss and Fazekas - 2005 - Group theory and octupolar order in $mathrm U m.pdf} + doi = {10/gf5c39}, + author = {Choi, J. and Ivashko, O. and Dennler, N. and Aoki, D. and {von Arx}, K. and Gerber, S. and Gutowski, O. and Fischer, M. H. and Strempfer, J. and {v. Zimmermann}, M. and Chang, J.}, + month = dec, + year = {2018}, + pages = {241113}, + file = {/home/pants/.zotero/data/storage/8IBGVH7U/Choi et al. - 2018 - Pressure-induced rotational symmetry breaking in $.pdf} } -@article{haule_arrested_2009, - title = {Arrested {{Kondo}} Effect and Hidden Order in {{URu}}{\textsubscript{2}}{{Si}}{\textsubscript{2}}}, - volume = {5}, - issn = {1745-2481}, - abstract = {Complex electronic matter shows subtle forms of self-organization, which are almost invisible to the available experimental tools. One prominent example is provided by the heavy-fermion material URu2Si2. At high temperature, the 5f electrons of uranium carry a very large entropy. This entropy is released at 17.5 K by means of a second-order phase transition1 to a state that remains shrouded in mystery, termed a `hidden order' state2. Here, we develop a first-principles theoretical method to analyse the electronic spectrum of correlated materials as a function of the position inside the unit cell of the crystal and use it to identify the low-energy excitations of URu2Si2. We identify the order parameter of the hidden-order state and show that it is intimately connected to magnetism. Below 70 K, the 5f electrons undergo a multichannel Kondo effect, which is `arrested' at low temperature by the crystal-field splitting. At lower temperatures, two broken-symmetry states emerge, characterized by a complex order parameter {$\psi$}. A real {$\psi$} describes the hidden-order phase and an imaginary {$\psi$} corresponds to the large-moment antiferromagnetic phase. Together, they provide a unified picture of the two broken-symmetry phases in this material.}, - language = {en}, +@article{hassinger_temperature-pressure_2008, + title = {Temperature-Pressure Phase Diagram of \$\textbackslash{}mathrm\{\vphantom\}{{U}}\vphantom\{\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$ from Resistivity Measurements and Ac Calorimetry: {{Hidden}} Order and {{Fermi}}-Surface Nesting}, + volume = {77}, + shorttitle = {Temperature-Pressure Phase Diagram of \$\textbackslash{}mathrm\{\vphantom\}{{U}}\vphantom\{\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$ from Resistivity Measurements and Ac Calorimetry}, + abstract = {By performing combined resistivity and calorimetric experiments under pressure, we have determined a precise temperature-pressure (T,P) phase diagram of the heavy fermion compound URu2Si2. It will be compared with previous diagrams determined by elastic neutron diffraction and strain gauge techniques. At first glance, the low-pressure ordered phase referred to as hidden order is dominated by Fermi-surface nesting, which has strong consequences on the localized spin dynamics. The high-pressure phase is dominated by large moment antiferromagnetism (LMAF) coexisting with at least dynamical nesting needed to restore on cooling a local moment behavior. ac calorimetry confirms unambiguously that bulk superconductivity does not coexist with LMAF. URu2Si2 is one of the most spectacular examples of the dual itinerant and local character of uranium-based heavy fermion compounds.}, number = {11}, - journal = {Nature Physics}, - doi = {10/fw2wcx}, - author = {Haule, Kristjan and Kotliar, Gabriel}, - month = nov, - year = {2009}, - pages = {796-799}, - file = {/home/pants/Zotero/storage/L3WFEVLT/Haule and Kotliar - 2009 - Arrested Kondo effect and hidden order in URusub.pdf} + journal = {Physical Review B}, + doi = {10.1103/physrevb.77.115117}, + author = {Hassinger, E. and Knebel, G. and Izawa, K. and Lejay, P. and Salce, B. and Flouquet, J.}, + month = mar, + year = {2008}, + pages = {115117}, + file = {/home/pants/.zotero/data/storage/U5V8JT6U/Hassinger et al. - 2008 - Temperature-pressure phase diagram of $mathrm U .pdf} } @article{kambe_odd-parity_2018, @@ -164,7 +149,23 @@ month = jun, year = {2018}, pages = {235142}, - file = {/home/pants/Zotero/storage/UQWWD3SU/Kambe et al. - 2018 - Odd-parity electronic multipolar ordering in $ ma.pdf} + file = {/home/pants/.zotero/data/storage/UQWWD3SU/Kambe et al. - 2018 - Odd-parity electronic multipolar ordering in $ ma.pdf} +} + +@article{haule_arrested_2009, + title = {Arrested {{Kondo}} Effect and Hidden Order in {{URu}}{\textsubscript{2}}{{Si}}{\textsubscript{2}}}, + volume = {5}, + issn = {1745-2481}, + abstract = {Complex electronic matter shows subtle forms of self-organization, which are almost invisible to the available experimental tools. One prominent example is provided by the heavy-fermion material URu2Si2. At high temperature, the 5f electrons of uranium carry a very large entropy. This entropy is released at 17.5 K by means of a second-order phase transition1 to a state that remains shrouded in mystery, termed a `hidden order' state2. Here, we develop a first-principles theoretical method to analyse the electronic spectrum of correlated materials as a function of the position inside the unit cell of the crystal and use it to identify the low-energy excitations of URu2Si2. We identify the order parameter of the hidden-order state and show that it is intimately connected to magnetism. Below 70 K, the 5f electrons undergo a multichannel Kondo effect, which is `arrested' at low temperature by the crystal-field splitting. At lower temperatures, two broken-symmetry states emerge, characterized by a complex order parameter {$\psi$}. A real {$\psi$} describes the hidden-order phase and an imaginary {$\psi$} corresponds to the large-moment antiferromagnetic phase. Together, they provide a unified picture of the two broken-symmetry phases in this material.}, + language = {en}, + number = {11}, + journal = {Nature Physics}, + doi = {10/fw2wcx}, + author = {Haule, Kristjan and Kotliar, Gabriel}, + month = nov, + year = {2009}, + pages = {796-799}, + file = {/home/pants/.zotero/data/storage/L3WFEVLT/Haule and Kotliar - 2009 - Arrested Kondo effect and hidden order in URusub.pdf} } @article{kusunose_hidden_2011, @@ -179,7 +180,7 @@ month = jul, year = {2011}, pages = {084702}, - file = {/home/pants/Zotero/storage/VSG5VAMT/Kusunose and Harima - 2011 - On the Hidden Order in URu2Si2 – Antiferro Hexadec.pdf} + file = {/home/pants/.zotero/data/storage/VSG5VAMT/Kusunose and Harima - 2011 - On the Hidden Order in URu2Si2 – Antiferro Hexadec.pdf} } @article{kung_chirality_2015, @@ -201,10 +202,24 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = mar, year = {2015}, pages = {1339-1342}, - file = {/home/pants/Zotero/storage/E93SDWTG/Kung et al. - 2015 - Chirality density wave of the “hidden order” phase.pdf}, + file = {/home/pants/.zotero/data/storage/E93SDWTG/Kung et al. - 2015 - Chirality density wave of the “hidden order” phase.pdf}, pmid = {25678557} } +@article{cricchio_itinerant_2009, + title = {Itinerant {{Magnetic Multipole Moments}} of {{Rank Five}} as the {{Hidden Order}} in \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, + volume = {103}, + abstract = {A broken symmetry ground state without any magnetic moments has been calculated by means of the local-density approximation to density functional theory plus a local exchange term, the so-called LDA+U approach, for URu2Si2. The solution is analyzed in terms of a multipole tensor expansion of the itinerant density matrix and is found to be a nontrivial magnetic multipole. Analysis and further calculations show that this type of multipole enters naturally in time reversal breaking in the presence of large effective spin-orbit coupling and coexists with magnetic moments for most magnetic actinides.}, + number = {10}, + journal = {Physical Review Letters}, + doi = {10/csgzd4}, + author = {Cricchio, Francesco and Bultmark, Fredrik and Gr{\aa}n{\"a}s, Oscar and Nordstr{\"o}m, Lars}, + month = sep, + year = {2009}, + pages = {107202}, + file = {/home/pants/.zotero/data/storage/KAXQ32EJ/Cricchio et al. - 2009 - Itinerant Magnetic Multipole Moments of Rank Five .pdf} +} + @article{ohkawa_quadrupole_1999, title = {Quadrupole and Dipole Orders in {{URu2Si2}}}, volume = {11}, @@ -218,21 +233,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = nov, year = {1999}, pages = {L519--L524}, - file = {/home/pants/Zotero/storage/AG7SQ5WT/Ohkawa and Shimizu - 1999 - Quadrupole and dipole orders in URu2Si2.pdf} -} - -@article{cricchio_itinerant_2009, - title = {Itinerant {{Magnetic Multipole Moments}} of {{Rank Five}} as the {{Hidden Order}} in \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, - volume = {103}, - abstract = {A broken symmetry ground state without any magnetic moments has been calculated by means of the local-density approximation to density functional theory plus a local exchange term, the so-called LDA+U approach, for URu2Si2. The solution is analyzed in terms of a multipole tensor expansion of the itinerant density matrix and is found to be a nontrivial magnetic multipole. Analysis and further calculations show that this type of multipole enters naturally in time reversal breaking in the presence of large effective spin-orbit coupling and coexists with magnetic moments for most magnetic actinides.}, - number = {10}, - journal = {Physical Review Letters}, - doi = {10/csgzd4}, - author = {Cricchio, Francesco and Bultmark, Fredrik and Gr{\aa}n{\"a}s, Oscar and Nordstr{\"o}m, Lars}, - month = sep, - year = {2009}, - pages = {107202}, - file = {/home/pants/Zotero/storage/KAXQ32EJ/Cricchio et al. - 2009 - Itinerant Magnetic Multipole Moments of Rank Five .pdf} + file = {/home/pants/.zotero/data/storage/AG7SQ5WT/Ohkawa and Shimizu - 1999 - Quadrupole and dipole orders in URu2Si2.pdf} } @article{santini_crystal_1994, @@ -246,7 +247,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = aug, year = {1994}, pages = {1027-1030}, - file = {/home/pants/Zotero/storage/2ZPUF4NZ/Santini and Amoretti - 1994 - Crystal Field Model of the Magnetic Properties of .pdf} + file = {/home/pants/.zotero/data/storage/2ZPUF4NZ/Santini and Amoretti - 1994 - Crystal Field Model of the Magnetic Properties of .pdf} } @article{harima_why_2010, @@ -261,7 +262,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = mar, year = {2010}, pages = {033705}, - file = {/home/pants/Zotero/storage/2MY7VK9P/Harima et al. - 2010 - Why the Hidden Order in URu2Si2 Is Still Hidden–On.pdf} + file = {/home/pants/.zotero/data/storage/2MY7VK9P/Harima et al. - 2010 - Why the Hidden Order in URu2Si2 Is Still Hidden–On.pdf} } @article{thalmeier_signatures_2011, @@ -275,7 +276,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = apr, year = {2011}, pages = {165110}, - file = {/home/pants/Zotero/storage/UD5E2IUD/Thalmeier and Takimoto - 2011 - Signatures of hidden-order symmetry in torque osci.pdf} + file = {/home/pants/.zotero/data/storage/UD5E2IUD/Thalmeier and Takimoto - 2011 - Signatures of hidden-order symmetry in torque osci.pdf} } @article{tonegawa_cyclotron_2012, @@ -289,7 +290,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = jul, year = {2012}, pages = {036401}, - file = {/home/pants/Zotero/storage/MHBZ6QTK/Tonegawa et al. - 2012 - Cyclotron Resonance in the Hidden-Order Phase of $.pdf} + file = {/home/pants/.zotero/data/storage/MHBZ6QTK/Tonegawa et al. - 2012 - Cyclotron Resonance in the Hidden-Order Phase of $.pdf} } @article{rau_hidden_2012, @@ -303,7 +304,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = jun, year = {2012}, pages = {245112}, - file = {/home/pants/Zotero/storage/6HP8DPHU/Rau and Kee - 2012 - Hidden and antiferromagnetic order as a rank-5 sup.pdf} + file = {/home/pants/.zotero/data/storage/6HP8DPHU/Rau and Kee - 2012 - Hidden and antiferromagnetic order as a rank-5 sup.pdf} } @article{riggs_evidence_2015, @@ -318,7 +319,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = mar, year = {2015}, pages = {6425}, - file = {/home/pants/Zotero/storage/Z57IE8J9/Riggs et al. - 2015 - Evidence for a nematic component to the hidden-ord.pdf} + file = {/home/pants/.zotero/data/storage/Z57IE8J9/Riggs et al. - 2015 - Evidence for a nematic component to the hidden-ord.pdf} } @article{hoshino_resolution_2013, @@ -333,7 +334,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = mar, year = {2013}, pages = {044707}, - file = {/home/pants/Zotero/storage/TY637XGC/Hoshino et al. - 2013 - Resolution of Entropy (lnsqrt 2 ) by Ordering .pdf} + file = {/home/pants/.zotero/data/storage/TY637XGC/Hoshino et al. - 2013 - Resolution of Entropy (lnsqrt 2 ) by Ordering .pdf} } @article{ikeda_theory_1998, @@ -348,7 +349,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = oct, year = {1998}, pages = {3723-3726}, - file = {/home/pants/Zotero/storage/QNE8NK4Q/Ikeda and Ohashi - 1998 - Theory of Unconventional Spin Density Wave A Poss.pdf} + file = {/home/pants/.zotero/data/storage/QNE8NK4Q/Ikeda and Ohashi - 1998 - Theory of Unconventional Spin Density Wave A Poss.pdf} } @article{chandra_hastatic_2013, @@ -364,34 +365,7 @@ Raman spectroscopy is used to uncover an unusual ordering in the low-temperature month = jan, year = {2013}, pages = {621-626}, - file = {/home/pants/Zotero/storage/B272KFL9/Chandra et al. - 2013 - Hastatic order in the heavy-fermion compound URus.pdf} -} - -@article{harrison_hidden_2019, - title = {Hidden Valence Transition in {{URu2Si2}}?}, - abstract = {The term "hidden order" refers to an as yet unidentified form of -broken-symmetry order parameter that is presumed to exist in the strongly -correlated electron system URu2Si2 on the basis of the reported similarity of -the heat capacity at its phase transition at To\textasciitilde{}17 K to that produced by -Bardeen-Cooper-Schrieffer (BCS) mean field theory. Here we show that the phase -boundary in URu2Si2 has the elliptical form expected for an entropy-driven -phase transition, as has been shown to accompany a change in valence. We show -one characteristic feature of such a transition is that the ratio of the -critical magnetic field to the critical temperature is defined solely in terms -of the effective quasiparticle g-factor, which we find to be in quantitative -agreement with prior g-factor measurements. We further find the anomaly in the -heat capacity at To to be significantly sharper than a BCS phase transition, -and, once quasiparticle excitations across the hybridization gap are taken into -consideration, loses its resemblance to a second order phase transition. Our -findings imply that a change in valence dominates the thermodynamics of the -phase boundary in URu2Si2, and eclipses any significant contribution to the -thermodynamics from a hidden order parameter.}, - language = {en}, - author = {Harrison, Neil and Jaime, Marcelo}, - month = feb, - year = {2019}, - keywords = {⛔ No DOI found}, - file = {/home/pants/Zotero/storage/39JAA4F8/Harrison and Jaime - 2019 - Hidden valence transition in URu2Si2.pdf} + file = {/home/pants/.zotero/data/storage/B272KFL9/Chandra et al. - 2013 - Hastatic order in the heavy-fermion compound URus.pdf} } @article{ikeda_emergent_2012, @@ -407,66 +381,82 @@ thermodynamics from a hidden order parameter.}, month = jul, year = {2012}, pages = {528-533}, - file = {/home/pants/Zotero/storage/9NYNGB45/Ikeda et al. - 2012 - Emergent rank-5 nematic order in URusub2subSi.pdf} + file = {/home/pants/.zotero/data/storage/9NYNGB45/Ikeda et al. - 2012 - Emergent rank-5 nematic order in URusub2subSi.pdf} } -@article{ghosh_single-component_2019, - archivePrefix = {arXiv}, - eprinttype = {arxiv}, - eprint = {1903.00552}, - primaryClass = {cond-mat, physics:physics}, - title = {Single-{{Component Order Parameter}} in {{URu}}\$\_2\${{Si}}\$\_2\$ {{Uncovered}} by {{Resonant Ultrasound Spectroscopy}} and {{Machine Learning}}}, - abstract = {URu\$\_2\$Si\$\_2\$ exhibits a clear phase transition at T\$\_\{HO\}= 17.5\textasciitilde\$K to a low-temperature phase known as "hidden order" (HO). Even the most basic information needed to construct a theory of this state---such as the number of components in the order parameter---has been lacking. Here we use resonant ultrasound spectroscopy (RUS) and machine learning to determine that the order parameter of HO is one-dimensional (singlet), ruling out a large class of theories based on two-dimensional (doublet) order parameters. This strict constraint is independent of any microscopic mechanism, and independent of other symmetries that HO may break. Our technique is general for second-order phase transitions, and can discriminate between nematic (singlet) versus loop current (doublet) order in the high-\textbackslash{}Tc cuprates, and conventional (singlet) versus the proposed \$p\_x+ip\_y\$ (doublet) superconductivity in Sr\$\_2\$RuO\$\_4\$. The machine learning framework we develop should be readily adaptable to other spectroscopic techniques where missing resonances confound traditional analysis, such as NMR.}, - journal = {arXiv:1903.00552 [cond-mat, physics:physics]}, - author = {Ghosh, Sayak and Matty, Michael and Baumbach, Ryan and Bauer, Eric D. and Modic, K. A. and Shekhter, Arkady and Mydosh, J. A. and Kim, Eun-Ah and Ramshaw, B. J.}, - month = mar, - year = {2019}, - keywords = {⛔ No DOI found,Condensed Matter - Strongly Correlated Electrons,Physics - Data Analysis; Statistics and Probability}, - file = {/home/pants/Zotero/storage/WV2NUDE9/Ghosh et al. - 2019 - Single-Component Order Parameter in URu$_2$Si$_2$ .pdf} +@article{kiss_group_2005, + title = {Group Theory and Octupolar Order in \$\textbackslash{}mathrm\{\vphantom\}{{U}}\vphantom\{\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, + volume = {71}, + abstract = {Recent experiments on URu2Si2URu2Si2 show that the low-pressure hidden order is nonmagnetic but it breaks time reversal invariance. Restricting our attention to local order parameters of 5f25f2 shells, we find that the best candidate for hidden order is staggered order of either Tz{$\beta$}T{$\beta$}z or TxyzTxyz octupoles. Group theoretical arguments for the effect of symmetry-lowering perturbations (magnetic field, mechanical stress) predict behavior in good overall agreement with observations. We illustrate our general arguments on the example of a five-state crystal field model which differs in several details from models discussed in the literature. The general appearance of the mean field phase diagram agrees with the experimental results. In particular, we find that (a) at zero magnetic field, there is a first-order phase boundary between octupolar order and large-moment antiferromagnetism with increasing hydrostatic pressure; (b) arbitrarily weak uniaxial pressure induces staggered magnetic moments in the octupolar phase; and (c) a new phase with different symmetry appears at large magnetic fields.}, + number = {5}, + journal = {Physical Review B}, + doi = {10.1103/physrevb.71.054415}, + author = {Kiss, Annam{\'a}ria and Fazekas, Patrik}, + month = feb, + year = {2005}, + pages = {054415}, + file = {/home/pants/.zotero/data/storage/YTARVDIM/Kiss and Fazekas - 2005 - Group theory and octupolar order in $mathrm U m.pdf} } -@article{hassinger_temperature-pressure_2008, - title = {Temperature-Pressure Phase Diagram of \$\textbackslash{}mathrm\{\vphantom\}{{U}}\vphantom\{\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$ from Resistivity Measurements and Ac Calorimetry: {{Hidden}} Order and {{Fermi}}-Surface Nesting}, - volume = {77}, - shorttitle = {Temperature-Pressure Phase Diagram of \$\textbackslash{}mathrm\{\vphantom\}{{U}}\vphantom\{\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$ from Resistivity Measurements and Ac Calorimetry}, - abstract = {By performing combined resistivity and calorimetric experiments under pressure, we have determined a precise temperature-pressure (T,P) phase diagram of the heavy fermion compound URu2Si2. It will be compared with previous diagrams determined by elastic neutron diffraction and strain gauge techniques. At first glance, the low-pressure ordered phase referred to as hidden order is dominated by Fermi-surface nesting, which has strong consequences on the localized spin dynamics. The high-pressure phase is dominated by large moment antiferromagnetism (LMAF) coexisting with at least dynamical nesting needed to restore on cooling a local moment behavior. ac calorimetry confirms unambiguously that bulk superconductivity does not coexist with LMAF. URu2Si2 is one of the most spectacular examples of the dual itinerant and local character of uranium-based heavy fermion compounds.}, - number = {11}, +@article{varshni_temperature_1970, + title = {Temperature {{Dependence}} of the {{Elastic Constants}}}, + volume = {2}, + abstract = {The following two equations are proposed for the temperature dependence of the elastic stiffness constants: cij=c0ij-s(etT-1) and cij=a-bT2(T+c), where c0ij, s, t, a, b, and c are constants. The applicability of these two equations and that of Wachtman's equation is examined for 57 elastic constants of 22 substances. The first equation has a theoretical justification and gives the best over-all results. Neither of the three equations give the theoretically expected T4 dependence at low temperatures, and therefore they are not expected to give very accurate results at very low temperatures ({$\lessequivlnt\Theta$}D50). A new melting criterion is also examined.}, + number = {10}, journal = {Physical Review B}, - doi = {10.1103/physrevb.77.115117}, - author = {Hassinger, E. and Knebel, G. and Izawa, K. and Lejay, P. and Salce, B. and Flouquet, J.}, - month = mar, - year = {2008}, - pages = {115117}, - file = {/home/pants/Zotero/storage/U5V8JT6U/Hassinger et al. - 2008 - Temperature-pressure phase diagram of $mathrm U .pdf} + doi = {10.1103/physrevb.2.3952}, + author = {Varshni, Y. P.}, + month = nov, + year = {1970}, + pages = {3952-3958}, + file = {/home/pants/.zotero/data/storage/QN7TLJV7/Varshni - 1970 - Temperature Dependence of the Elastic Constants.pdf} } -@article{choi_pressure-induced_2018, - title = {Pressure-Induced Rotational Symmetry Breaking in \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, - volume = {98}, - abstract = {Phase transitions and symmetry are intimately linked. Melting of ice, for example, restores translation invariance. The mysterious hidden order (HO) phase of URu2Si2 has, despite relentless research efforts, kept its symmetry breaking element intangible. Here, we present a high-resolution x-ray diffraction study of the URu2Si2 crystal structure as a function of hydrostatic pressure. Below a critical pressure threshold pc{$\approx$}3 kbar, no tetragonal lattice symmetry breaking is observed even below the HO transition THO=17.5 K. For p{$>$}pc, however, a pressure-induced rotational symmetry breaking is identified with an onset temperatures TOR{$\sim$}100 K. The emergence of an orthorhombic phase is found and discussed in terms of an electronic nematic order that appears unrelated to the HO, but with possible relevance for the pressure-induced antiferromagnetic (AF) phase. Existing theories describe the HO and AF phases through an adiabatic continuity of a complex order parameter. Since none of these theories predicts a pressure-induced nematic order, our finding adds an additional symmetry breaking element to this long-standing problem.}, - number = {24}, - journal = {Physical Review B}, - doi = {10/gf5c39}, - author = {Choi, J. and Ivashko, O. and Dennler, N. and Aoki, D. and {von Arx}, K. and Gerber, S. and Gutowski, O. and Fischer, M. H. and Strempfer, J. and {v. Zimmermann}, M. and Chang, J.}, +@article{hornreich_critical_1975, + title = {Critical {{Behavior}} at the {{Onset}} of \$\textbackslash{}stackrel\{\textbackslash{}ensuremath\{\textbackslash{}rightarrow\}\}\{\textbackslash{}mathrm\{k\}\}\$-{{Space Instability}} on the \$\textbackslash{}ensuremath\{\textbackslash{}lambda\}\$ {{Line}}}, + volume = {35}, + abstract = {We calculate the critical behavior of systems having a multicritical point of a new type, hereafter called a Lifshitz point, which separates ordered phases with \textrightarrowk=0 and \textrightarrowk{$\not =$}0 along the {$\lambda$} line. For anisotropic systems, the correlation function is described in terms of four critical exponents, whereas for isotropic systems two exponents suffice. Critical exponents are calculated using an {$\epsilon$}-type expansion.}, + number = {25}, + journal = {Physical Review Letters}, + doi = {10.1103/PhysRevLett.35.1678}, + author = {Hornreich, R. M. and Luban, Marshall and Shtrikman, S.}, month = dec, - year = {2018}, - pages = {241113}, - file = {/home/pants/Zotero/storage/8IBGVH7U/Choi et al. - 2018 - Pressure-induced rotational symmetry breaking in $.pdf} + year = {1975}, + pages = {1678-1681}, + file = {/home/pants/.zotero/data/storage/GBYIESIW/Hornreich et al_1975_Critical Behavior at the Onset of.pdf;/home/pants/.zotero/data/storage/KBYQHWSH/PhysRevLett.35.html} } -@article{chandra_origin_2013, - title = {Origin of the {{Large Anisotropy}} in The\$\textbackslash{}upchi\${{3Anomaly inURu2Si2}}}, - volume = {449}, - issn = {1742-6596}, - abstract = {Motivated by recent quantum oscillations experiments on U Ru2Si2, we discuss the microscopic origin of the large anisotropy observed many years ago in the anomaly of the nonlinear susceptibility in this same material. We show that the magnitude of this anomaly emerges naturally from hastatic order, a proposal for hidden order that is a two-component spinor arising from the hybridization of a non-Kramers {$\Gamma$}5 doublet with Kramers conduction electrons. A prediction is made for the angular anisotropy of the nonlinear susceptibility anomaly as a test of this proposed order parameter for U Ru2Si2.}, +@article{inoue_high-field_2001, + series = {Proceedings of the {{Sixth International}} {{Symposium}} on {{Research}} in {{High Magnetic Fields}}}, + title = {High-Field Magnetization of {{URu2Si2}} under High Pressure}, + volume = {294-295}, + issn = {0921-4526}, + abstract = {The temperature dependence of the magnetic susceptibility and the high-field magnetization up to 55T are measured for URu2Si2 under high pressures up to 1GPa. Both T{$\chi$}max and TN in the susceptibility increase with increasing pressure. The value of the susceptibility below T{$\chi$}max decreases with increasing pressure. The three high-field metamagnetic transitions at Hc1=35.1T, Hc2=36.5T and Hc3=39.6T at ambient pressure, show different pressure-dependent behaviors. The metamagnetic transition at Hc1 broadens but survives and its transition field increases with increasing pressure. However, the transition at Hc2 is smeared out and disappears above 0.4GPa. The transition at Hc3 broadens more clearly than the transition at Hc1. The fact that both T{$\chi$}max and the metamagnetic transition fields increase suggests that the interaction between the f-electrons and the conduction electrons is enhanced by pressure.}, + journal = {Physica B: Condensed Matter}, + doi = {10.1016/S0921-4526(00)00657-8}, + author = {Inoue, T. and Kindo, K. and Okuni, H. and Sugiyama, K. and Haga, Y. and Yamamoto, E. and Kobayashi, T. C. and Uwatoko, Y. and Onuki, Y.}, + month = jan, + year = {2001}, + keywords = {High pressure,High-field magnetization,Metamagnetic transition,URuSi}, + pages = {271-275}, + file = {/home/pants/.zotero/data/storage/CDTQB6PI/Inoue et al_2001_High-field magnetization of URu2Si2 under high pressure.pdf;/home/pants/.zotero/data/storage/323PS9NS/S0921452600006578.html} +} + +@article{shekhter_bounding_2013, + title = {Bounding the Pseudogap with a Line of Phase Transitions in {{YBa}}{\textsubscript{2}}{{Cu}}{\textsubscript{3}}{{O}}{\textsubscript{6+{\emph{{$\delta$} }}}}}, + volume = {498}, + copyright = {2013 Nature Publishing Group}, + issn = {1476-4687}, + abstract = {Close to optimal doping, the copper oxide superconductors show `strange metal' behaviour1,2, suggestive of strong fluctuations associated with a quantum critical point3,4,5,6. Such a critical point requires a line of classical phase transitions terminating at zero temperature near optimal doping inside the superconducting `dome'. The underdoped region of the temperature\textendash{}doping phase diagram from which superconductivity emerges is referred to as the `pseudogap'7,8,9,10,11,12,13 because evidence exists for partial gapping of the conduction electrons, but so far there is no compelling thermodynamic evidence as to whether the pseudogap is a distinct phase or a continuous evolution of physical properties on cooling. Here we report that the pseudogap in YBa2Cu3O6+{$\delta$} is a distinct phase, bounded by a line of phase transitions. The doping dependence of this line is such that it terminates at zero temperature inside the superconducting dome. From this we conclude that quantum criticality drives the strange metallic behaviour and therefore superconductivity in the copper oxide superconductors.}, language = {en}, - journal = {Journal of Physics: Conference Series}, - doi = {10.1088/1742-6596/449/1/012026}, - author = {Chandra, P. and Coleman, P. and Flint, R.}, - month = jul, + number = {7452}, + journal = {Nature}, + doi = {10.1038/nature12165}, + author = {Shekhter, Arkady and Ramshaw, B. J. and Liang, Ruixing and Hardy, W. N. and Bonn, D. A. and Balakirev, Fedor F. and McDonald, Ross D. and Betts, Jon B. and Riggs, Scott C. and Migliori, Albert}, + month = jun, year = {2013}, - pages = {012026}, - file = {/home/pants/Zotero/storage/7K2FNND4/Chandra et al. - 2013 - Origin of the Large Anisotropy in the$upchi$3Anom.pdf} + pages = {75-77}, + file = {/home/pants/.zotero/data/storage/Y3X6VXIK/Shekhter et al_2013_Bounding the pseudogap with a line of phase transitions in.pdf;/home/pants/.zotero/data/storage/ZZ3MR77N/nature12165.html} } @article{meng_imaging_2013, @@ -480,71 +470,82 @@ thermodynamics from a hidden order parameter.}, month = sep, year = {2013}, pages = {127002}, - file = {/home/pants/Zotero/storage/EBTUZTN7/Meng et al_2013_Imaging the Three-Dimensional Fermi-Surface Pairing near the Hidden-Order.pdf;/home/pants/Zotero/storage/U2Z93ZIJ/PhysRevLett.111.html} + file = {/home/pants/.zotero/data/storage/EBTUZTN7/Meng et al_2013_Imaging the Three-Dimensional Fermi-Surface Pairing near the Hidden-Order.pdf;/home/pants/.zotero/data/storage/U2Z93ZIJ/PhysRevLett.111.html} } -@article{shekhter_bounding_2013, - title = {Bounding the Pseudogap with a Line of Phase Transitions in {{YBa}}{\textsubscript{2}}{{Cu}}{\textsubscript{3}}{{O}}{\textsubscript{6+{\emph{{$\delta$} }}}}}, - volume = {498}, - copyright = {2013 Nature Publishing Group}, - issn = {1476-4687}, - abstract = {Close to optimal doping, the copper oxide superconductors show `strange metal' behaviour1,2, suggestive of strong fluctuations associated with a quantum critical point3,4,5,6. Such a critical point requires a line of classical phase transitions terminating at zero temperature near optimal doping inside the superconducting `dome'. The underdoped region of the temperature\textendash{}doping phase diagram from which superconductivity emerges is referred to as the `pseudogap'7,8,9,10,11,12,13 because evidence exists for partial gapping of the conduction electrons, but so far there is no compelling thermodynamic evidence as to whether the pseudogap is a distinct phase or a continuous evolution of physical properties on cooling. Here we report that the pseudogap in YBa2Cu3O6+{$\delta$} is a distinct phase, bounded by a line of phase transitions. The doping dependence of this line is such that it terminates at zero temperature inside the superconducting dome. From this we conclude that quantum criticality drives the strange metallic behaviour and therefore superconductivity in the copper oxide superconductors.}, +@article{chandra_origin_2013, + title = {Origin of the {{Large Anisotropy}} in The\$\textbackslash{}upchi\${{3Anomaly inURu2Si2}}}, + volume = {449}, + issn = {1742-6596}, + abstract = {Motivated by recent quantum oscillations experiments on U Ru2Si2, we discuss the microscopic origin of the large anisotropy observed many years ago in the anomaly of the nonlinear susceptibility in this same material. We show that the magnitude of this anomaly emerges naturally from hastatic order, a proposal for hidden order that is a two-component spinor arising from the hybridization of a non-Kramers {$\Gamma$}5 doublet with Kramers conduction electrons. A prediction is made for the angular anisotropy of the nonlinear susceptibility anomaly as a test of this proposed order parameter for U Ru2Si2.}, language = {en}, - number = {7452}, - journal = {Nature}, - doi = {10.1038/nature12165}, - author = {Shekhter, Arkady and Ramshaw, B. J. and Liang, Ruixing and Hardy, W. N. and Bonn, D. A. and Balakirev, Fedor F. and McDonald, Ross D. and Betts, Jon B. and Riggs, Scott C. and Migliori, Albert}, - month = jun, + journal = {Journal of Physics: Conference Series}, + doi = {10.1088/1742-6596/449/1/012026}, + author = {Chandra, P. and Coleman, P. and Flint, R.}, + month = jul, year = {2013}, - pages = {75-77}, - file = {/home/pants/Zotero/storage/Y3X6VXIK/Shekhter et al_2013_Bounding the pseudogap with a line of phase transitions in.pdf;/home/pants/Zotero/storage/ZZ3MR77N/nature12165.html} + pages = {012026}, + file = {/home/pants/.zotero/data/storage/7K2FNND4/Chandra et al. - 2013 - Origin of the Large Anisotropy in the$upchi$3Anom.pdf} } -@article{inoue_high-field_2001, - series = {Proceedings of the {{Sixth International}} {{Symposium}} on {{Research}} in {{High Magnetic Fields}}}, - title = {High-Field Magnetization of {{URu2Si2}} under High Pressure}, - volume = {294-295}, - issn = {0921-4526}, - abstract = {The temperature dependence of the magnetic susceptibility and the high-field magnetization up to 55T are measured for URu2Si2 under high pressures up to 1GPa. Both T{$\chi$}max and TN in the susceptibility increase with increasing pressure. The value of the susceptibility below T{$\chi$}max decreases with increasing pressure. The three high-field metamagnetic transitions at Hc1=35.1T, Hc2=36.5T and Hc3=39.6T at ambient pressure, show different pressure-dependent behaviors. The metamagnetic transition at Hc1 broadens but survives and its transition field increases with increasing pressure. However, the transition at Hc2 is smeared out and disappears above 0.4GPa. The transition at Hc3 broadens more clearly than the transition at Hc1. The fact that both T{$\chi$}max and the metamagnetic transition fields increase suggests that the interaction between the f-electrons and the conduction electrons is enhanced by pressure.}, - journal = {Physica B: Condensed Matter}, - doi = {10.1016/S0921-4526(00)00657-8}, - author = {Inoue, T. and Kindo, K. and Okuni, H. and Sugiyama, K. and Haga, Y. and Yamamoto, E. and Kobayashi, T. C. and Uwatoko, Y. and Onuki, Y.}, +@article{garel_commensurability_1976, + title = {Commensurability Effects on the Critical Behaviour of Systems with Helical Ordering}, + volume = {9}, + issn = {0022-3719}, + abstract = {The critical behaviour of an m-component spin system with helical ordering is studied using the renormalization group method to order epsilon 2 (where epsilon =4-d). For m=1 and 2 the system is equivalent to a 2m-vector model. For m=3 a first-order transition is expected. The effect of the commensurability of the helical structure with the lattice has been considered and is shown in certain situations to change the order of the transition.}, + language = {en}, + number = {10}, + journal = {Journal of Physics C: Solid State Physics}, + doi = {2011031909475300}, + author = {Garel, T. and Pfeuty, P.}, + month = may, + year = {1976}, + pages = {L245--L249}, + file = {/home/pants/.zotero/data/storage/34KTXA6I/Garel_Pfeuty_1976_Commensurability effects on the critical behaviour of systems with helical.pdf} +} + +@article{nicoll_onset_1977, + title = {Onset of Helical Order}, + volume = {86-88}, + issn = {0378-4363}, + abstract = {Renormalization group methods are used to describe systems which model critical phenomena at the onset of helical order. This onset is marked by a change in the ``bare propagator'' used in perturbation theory from a k2-dependence to a more general form. We consider systems which in the non-helical region exhibit O simultaneously critical phases. Results are given to first order in an {$\epsilon$}-expansion. For the isotropic case of k2L dependence and O = 2, we give {$\eta$} to first order in 1/n for d- {$\leqslant$} d {$\leqslant$} d+ where d+- are upper and lower borderline dimensions.}, + journal = {Physica B+C}, + doi = {10.1016/0378-4363(77)90620-9}, + author = {Nicoll, J. F. and Tuthill, G. F. and Chang, T. S. and Stanley, H. E.}, month = jan, - year = {2001}, - keywords = {High pressure,High-field magnetization,Metamagnetic transition,URuSi}, - pages = {271-275}, - file = {/home/pants/Zotero/storage/CDTQB6PI/Inoue et al_2001_High-field magnetization of URu2Si2 under high pressure.pdf;/home/pants/Zotero/storage/323PS9NS/S0921452600006578.html} + year = {1977}, + pages = {618-620}, + file = {/home/pants/.zotero/data/storage/ZLV5YFH6/Nicoll et al_1977_Onset of helical order.pdf;/home/pants/.zotero/data/storage/84ZZT6CN/0378436377906209.html} } -@article{hornreich_critical_1975, - title = {Critical {{Behavior}} at the {{Onset}} of \$\textbackslash{}stackrel\{\textbackslash{}ensuremath\{\textbackslash{}rightarrow\}\}\{\textbackslash{}mathrm\{k\}\}\$-{{Space Instability}} on the \$\textbackslash{}ensuremath\{\textbackslash{}lambda\}\$ {{Line}}}, - volume = {35}, - abstract = {We calculate the critical behavior of systems having a multicritical point of a new type, hereafter called a Lifshitz point, which separates ordered phases with \textrightarrowk=0 and \textrightarrowk{$\not =$}0 along the {$\lambda$} line. For anisotropic systems, the correlation function is described in terms of four critical exponents, whereas for isotropic systems two exponents suffice. Critical exponents are calculated using an {$\epsilon$}-type expansion.}, - number = {25}, - journal = {Physical Review Letters}, - doi = {10.1103/PhysRevLett.35.1678}, - author = {Hornreich, R. M. and Luban, Marshall and Shtrikman, S.}, - month = dec, - year = {1975}, - pages = {1678-1681}, - file = {/home/pants/Zotero/storage/GBYIESIW/Hornreich et al_1975_Critical Behavior at the Onset of.pdf;/home/pants/Zotero/storage/KBYQHWSH/PhysRevLett.35.html} +@article{nicoll_renormalization_1976, + title = {Renormalization Group Calculation for Critical Points of Higher Order with General Propagator}, + volume = {58}, + issn = {0375-9601}, + abstract = {We give first order perturbation results for the critical point exponents at order O critical points with anisotropic propagators. The exponent {$\eta$} is calculated to second order for isotropic propagators, and all O; 1/n expansion results are given for O = 2.}, + number = {1}, + journal = {Physics Letters A}, + doi = {10.1016/0375-9601(76)90527-2}, + author = {Nicoll, J. F. and Tuthill, G. F. and Chang, T. S. and Stanley, H. E.}, + month = jul, + year = {1976}, + pages = {1-2}, + file = {/home/pants/.zotero/data/storage/55AS69UD/Nicoll et al_1976_Renormalization group calculation for critical points of higher order with.pdf;/home/pants/.zotero/data/storage/L6WH4D36/0375960176905272.html} } -@article{selke_monte_1978, - title = {Monte Carlo Calculations near a Uniaxial {{Lifshitz}} Point}, - volume = {29}, - issn = {1431-584X}, - abstract = {The Monte Carlo method is applied to a threedimensional Ising model with nearest neighbour ferromagnetic interactions and next nearest neighbour antiferromagnetic interactions along one axis only. Special emphasis is given to the critical behaviour near the Lifshitz point.}, - language = {en}, +@article{hornreich_exactly_1977, + title = {Exactly Solvable Model Exhibiting a Multicritical Point}, + volume = {86}, + issn = {0378-4371}, + abstract = {A hypercubic d-dimensional lattice of spins with nearest neighbor ferromagnetic coupling and next nearest neighbor antiferromagnetic coupling along a single axis is studied in the spherical model limit (n\textrightarrow{$\infty$}) and is found to exhibit a multicritical point of the uniaxial Lifshitz type. The shape of the {$\lambda$} line is calculated explicitly in the vicinity of the multicritical point, and analytic expressions are given for the shift exponent {$\psi$}(d) and its amplitudes A{$\pm$}(d). The amplitude A\_(d) changes sign for d = 3.}, number = {2}, - journal = {Zeitschrift f{\"u}r Physik B Condensed Matter}, - doi = {10.1007/BF01313198}, - author = {Selke, Walter}, - month = jun, - year = {1978}, - keywords = {Complex System,Neural Network,Spectroscopy,State Physics,Monte Carlo Method}, - pages = {133-137}, - file = {/home/pants/Zotero/storage/5NRZEWP8/Selke_1978_Monte carlo calculations near a uniaxial Lifshitz point.pdf} + journal = {Physica A: Statistical Mechanics and its Applications}, + doi = {10.1016/0378-4371(77)90042-5}, + author = {Hornreich, R. M. and Luban, Marshall and Shtrikman, S.}, + month = feb, + year = {1977}, + pages = {465-470}, + file = {/home/pants/.zotero/data/storage/5MFN7M9Z/Hornreich et al_1977_Exactly solvable model exhibiting a multicritical point.pdf;/home/pants/.zotero/data/storage/CZNV72TI/0378437177900425.html} } @article{hornreich_critical_1975-1, @@ -559,67 +560,92 @@ thermodynamics from a hidden order parameter.}, month = dec, year = {1975}, pages = {269-270}, - file = {/home/pants/Zotero/storage/RED39SK4/Hornreich et al_1975_Critical exponents at a Lifshitz point to O(1-n).pdf;/home/pants/Zotero/storage/X8UJ5CHZ/037596017590465X.html} + file = {/home/pants/.zotero/data/storage/RED39SK4/Hornreich et al_1975_Critical exponents at a Lifshitz point to O(1-n).pdf;/home/pants/.zotero/data/storage/X8UJ5CHZ/037596017590465X.html} } -@article{hornreich_exactly_1977, - title = {Exactly Solvable Model Exhibiting a Multicritical Point}, - volume = {86}, - issn = {0378-4371}, - abstract = {A hypercubic d-dimensional lattice of spins with nearest neighbor ferromagnetic coupling and next nearest neighbor antiferromagnetic coupling along a single axis is studied in the spherical model limit (n\textrightarrow{$\infty$}) and is found to exhibit a multicritical point of the uniaxial Lifshitz type. The shape of the {$\lambda$} line is calculated explicitly in the vicinity of the multicritical point, and analytic expressions are given for the shift exponent {$\psi$}(d) and its amplitudes A{$\pm$}(d). The amplitude A\_(d) changes sign for d = 3.}, +@article{selke_monte_1978, + title = {Monte Carlo Calculations near a Uniaxial {{Lifshitz}} Point}, + volume = {29}, + issn = {1431-584X}, + abstract = {The Monte Carlo method is applied to a threedimensional Ising model with nearest neighbour ferromagnetic interactions and next nearest neighbour antiferromagnetic interactions along one axis only. Special emphasis is given to the critical behaviour near the Lifshitz point.}, + language = {en}, number = {2}, - journal = {Physica A: Statistical Mechanics and its Applications}, - doi = {10.1016/0378-4371(77)90042-5}, - author = {Hornreich, R. M. and Luban, Marshall and Shtrikman, S.}, - month = feb, - year = {1977}, - pages = {465-470}, - file = {/home/pants/Zotero/storage/5MFN7M9Z/Hornreich et al_1977_Exactly solvable model exhibiting a multicritical point.pdf;/home/pants/Zotero/storage/CZNV72TI/0378437177900425.html} + journal = {Zeitschrift f{\"u}r Physik B Condensed Matter}, + doi = {10.1007/BF01313198}, + author = {Selke, Walter}, + month = jun, + year = {1978}, + keywords = {Complex System,Neural Network,Spectroscopy,State Physics,Monte Carlo Method}, + pages = {133-137}, + file = {/home/pants/.zotero/data/storage/5NRZEWP8/Selke_1978_Monte carlo calculations near a uniaxial Lifshitz point.pdf} } -@article{nicoll_renormalization_1976, - title = {Renormalization Group Calculation for Critical Points of Higher Order with General Propagator}, - volume = {58}, - issn = {0375-9601}, - abstract = {We give first order perturbation results for the critical point exponents at order O critical points with anisotropic propagators. The exponent {$\eta$} is calculated to second order for isotropic propagators, and all O; 1/n expansion results are given for O = 2.}, - number = {1}, - journal = {Physics Letters A}, - doi = {10.1016/0375-9601(76)90527-2}, - author = {Nicoll, J. F. and Tuthill, G. F. and Chang, T. S. and Stanley, H. E.}, - month = jul, - year = {1976}, - pages = {1-2}, - file = {/home/pants/Zotero/storage/55AS69UD/Nicoll et al_1976_Renormalization group calculation for critical points of higher order with.pdf;/home/pants/Zotero/storage/L6WH4D36/0375960176905272.html} +@article{harrison_hidden_nodate, + archivePrefix = {arXiv}, + eprinttype = {arxiv}, + eprint = {1902.06588}, + title = {Hidden Valence Transition in {{URu2Si2}}?}, + abstract = {The term "hidden order" refers to an as yet unidentified form of broken-symmetry order parameter that is presumed to exist in the strongly correlated electron system URu2Si2 on the basis of the reported similarity of the heat capacity at its phase transition at To\textasciitilde{}17 K to that produced by Bardeen-Cooper-Schrieffer (BCS) mean field theory. Here we show that the phase boundary in URu2Si2 has the elliptical form expected for an entropy-driven phase transition, as has been shown to accompany a change in valence. We show one characteristic feature of such a transition is that the ratio of the critical magnetic field to the critical temperature is defined solely in terms of the effective quasiparticle g-factor, which we find to be in quantitative agreement with prior g-factor measurements. We further find the anomaly in the heat capacity at To to be significantly sharper than a BCS phase transition, and, once quasiparticle excitations across the hybridization gap are taken into consideration, loses its resemblance to a second order phase transition. Our findings imply that a change in valence dominates the thermodynamics of the phase boundary in URu2Si2, and eclipses any significant contribution to the thermodynamics from a hidden order parameter.}, + author = {Harrison, Neil and Jaime, Marcelo}, + keywords = {Condensed Matter - Strongly Correlated Electrons}, + file = {/home/pants/.zotero/data/storage/79NX4WI3/Harrison and Jaime - 2019 - Hidden valence transition in URu2Si2.pdf} } -@article{nicoll_onset_1977, - title = {Onset of Helical Order}, - volume = {86-88}, - issn = {0378-4363}, - abstract = {Renormalization group methods are used to describe systems which model critical phenomena at the onset of helical order. This onset is marked by a change in the ``bare propagator'' used in perturbation theory from a k2-dependence to a more general form. We consider systems which in the non-helical region exhibit O simultaneously critical phases. Results are given to first order in an {$\epsilon$}-expansion. For the isotropic case of k2L dependence and O = 2, we give {$\eta$} to first order in 1/n for d- {$\leqslant$} d {$\leqslant$} d+ where d+- are upper and lower borderline dimensions.}, - journal = {Physica B+C}, - doi = {10.1016/0378-4363(77)90620-9}, - author = {Nicoll, J. F. and Tuthill, G. F. and Chang, T. S. and Stanley, H. E.}, - month = jan, - year = {1977}, - pages = {618-620}, - file = {/home/pants/Zotero/storage/ZLV5YFH6/Nicoll et al_1977_Onset of helical order.pdf;/home/pants/Zotero/storage/84ZZT6CN/0378436377906209.html} +@article{ghosh_single-component_nodate, + archivePrefix = {arXiv}, + eprinttype = {arxiv}, + eprint = {1903.00552}, + title = {Single-{{Component Order Parameter}} in {{URu}}\$\_2\${{Si}}\$\_2\$ {{Uncovered}} by {{Resonant Ultrasound Spectroscopy}} and {{Machine Learning}}}, + abstract = {URu\$\_2\$Si\$\_2\$ exhibits a clear phase transition at T\$\_\{HO\}= 17.5\textasciitilde\$K to a low-temperature phase known as "hidden order" (HO). Even the most basic information needed to construct a theory of this state---such as the number of components in the order parameter---has been lacking. Here we use resonant ultrasound spectroscopy (RUS) and machine learning to determine that the order parameter of HO is one-dimensional (singlet), ruling out a large class of theories based on two-dimensional (doublet) order parameters. This strict constraint is independent of any microscopic mechanism, and independent of other symmetries that HO may break. Our technique is general for second-order phase transitions, and can discriminate between nematic (singlet) versus loop current (doublet) order in the high-\textbackslash{}Tc cuprates, and conventional (singlet) versus the proposed \$p\_x+ip\_y\$ (doublet) superconductivity in Sr\$\_2\$RuO\$\_4\$. The machine learning framework we develop should be readily adaptable to other spectroscopic techniques where missing resonances confound traditional analysis, such as NMR.}, + author = {Ghosh, Sayak and Matty, Michael and Baumbach, Ryan and Bauer, Eric D. and Modic, K. A. and Shekhter, Arkady and Mydosh, J. A. and Kim, Eun-Ah and Ramshaw, B. J.}, + keywords = {Condensed Matter - Strongly Correlated Electrons,Physics - Data Analysis; Statistics and Probability}, + file = {/home/pants/.zotero/data/storage/XIE9PPL6/Ghosh et al. - 2019 - Single-Component Order Parameter in URu$_2$Si$_2$ .pdf} } -@article{garel_commensurability_1976, - title = {Commensurability Effects on the Critical Behaviour of Systems with Helical Ordering}, - volume = {9}, - issn = {0022-3719}, - abstract = {The critical behaviour of an m-component spin system with helical ordering is studied using the renormalization group method to order epsilon 2 (where epsilon =4-d). For m=1 and 2 the system is equivalent to a 2m-vector model. For m=3 a first-order transition is expected. The effect of the commensurability of the helical structure with the lattice has been considered and is shown in certain situations to change the order of the transition.}, +@article{ramshaw_avoided_2015, + title = {Avoided Valence Transition in a Plutonium Superconductor}, + volume = {112}, + issn = {0027-8424, 1091-6490}, + abstract = {The d and f electrons in correlated metals are often neither fully localized around their host nuclei nor fully itinerant. This localized/itinerant duality underlies the correlated electronic states of the high-TcTc{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:msub{$><$}mml:mi{$>$}T{$<$}/mml:mi{$><$}mml:mi{$>$}c{$<$}/mml:mi{$><$}/mml:msub{$><$}/mml:mrow{$><$}/mml:math{$>$} cuprate superconductors and the heavy-fermion intermetallics and is nowhere more apparent than in the 5f5f{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:mn{$>$}5{$<$}/mml:mn{$><$}mml:mi{$>$}f{$<$}/mml:mi{$><$}/mml:mrow{$><$}/mml:math{$>$} valence electrons of plutonium. Here, we report the full set of symmetry-resolved elastic moduli of PuCoGa5\textemdash{}the highest TcTc{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:msub{$><$}mml:mi{$>$}T{$<$}/mml:mi{$><$}mml:mi{$>$}c{$<$}/mml:mi{$><$}/mml:msub{$><$}/mml:mrow{$><$}/mml:math{$>$} superconductor of the heavy fermions (TcTc{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:msub{$><$}mml:mi{$>$}T{$<$}/mml:mi{$><$}mml:mi{$>$}c{$<$}/mml:mi{$><$}/mml:msub{$><$}/mml:mrow{$><$}/mml:math{$>$} = 18.5 K)\textemdash{}and find that the bulk modulus softens anomalously over a wide range in temperature above TcTc{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:msub{$><$}mml:mi{$>$}T{$<$}/mml:mi{$><$}mml:mi{$>$}c{$<$}/mml:mi{$><$}/mml:msub{$><$}/mml:mrow{$><$}/mml:math{$>$}. The elastic symmetry channel in which this softening occurs is characteristic of a valence instability\textemdash{}therefore, we identify the elastic softening with fluctuations of the plutonium 5f mixed-valence state. These valence fluctuations disappear when the superconducting gap opens at TcTc{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:msub{$><$}mml:mi{$>$}T{$<$}/mml:mi{$><$}mml:mi{$>$}c{$<$}/mml:mi{$><$}/mml:msub{$><$}/mml:mrow{$><$}/mml:math{$>$}, suggesting that electrons near the Fermi surface play an essential role in the mixed-valence physics of this system and that PuCoGa5 avoids a valence transition by entering the superconducting state. The lack of magnetism in PuCoGa5 has made it difficult to reconcile with most other heavy-fermion superconductors, where superconductivity is generally believed to be mediated by magnetic fluctuations. Our observations suggest that valence fluctuations play a critical role in the unusually high TcTc{$<$}mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"{$><$}mml:mrow{$><$}mml:msub{$><$}mml:mi{$>$}T{$<$}/mml:mi{$><$}mml:mi{$>$}c{$<$}/mml:mi{$><$}/mml:msub{$><$}/mml:mrow{$><$}/mml:math{$>$} of PuCoGa5.}, language = {en}, - number = {10}, - journal = {Journal of Physics C: Solid State Physics}, - doi = {10.1088/0022-3719/9/10/001}, - author = {Garel, T. and Pfeuty, P.}, - month = may, - year = {1976}, - pages = {L245--L249}, - file = {/home/pants/Zotero/storage/34KTXA6I/Garel_Pfeuty_1976_Commensurability effects on the critical behaviour of systems with helical.pdf} + number = {11}, + journal = {Proceedings of the National Academy of Sciences}, + doi = {10.1073/pnas.1421174112}, + author = {Ramshaw, B. J. and Shekhter, Arkady and McDonald, Ross D. and Betts, Jon B. and Mitchell, J. N. and Tobash, P. H. and Mielke, C. H. and Bauer, E. D. and Migliori, Albert}, + month = mar, + year = {2015}, + keywords = {heavy fermions,quantum criticality,resonant ultrasound spectroscopy,unconventional superconductivity,valence fluctuations}, + pages = {3285-3289}, + file = {/home/pants/.zotero/data/storage/ERT8A25E/Ramshaw et al. - 2015 - Avoided valence transition in a plutonium supercon.pdf}, + pmid = {25737548} +} + +@article{luthi_sound_1970, + title = {Sound {{Propagation}} near the {{Structural Phase Transition}} in {{Strontium Titanate}}}, + volume = {2}, + abstract = {Finite ultrasonic velocity changes at the structural phase transition in SrTi03 are observed for different modes. They are interrelated and correlated by theory. No critical effects are observed.}, + number = {4}, + journal = {Physical Review B}, + doi = {10.1103/PhysRevB.2.1211}, + author = {L{\"u}thi, B. and Moran, T. J.}, + month = aug, + year = {1970}, + pages = {1211-1214}, + file = {/home/pants/.zotero/data/storage/RQZGTK9L/Lüthi and Moran - 1970 - Sound Propagation near the Structural Phase Transi.pdf} +} + +@article{bak_commensurate_1982, + title = {Commensurate Phases, Incommensurate Phases and the Devil's Staircase}, + volume = {45}, + issn = {0034-4885, 1361-6633}, + number = {6}, + journal = {Reports on Progress in Physics}, + doi = {10.1088/0034-4885/45/6/001}, + author = {Bak, P}, + month = jun, + year = {1982}, + pages = {587-629}, + file = {/home/pants/.zotero/data/storage/TYKMSDX7/Bak - 1982 - Commensurate phases, incommensurate phases and the.pdf} } @@ -6,9 +6,8 @@ % Our mysterious boy \def\urusi{URu$_{\text2}$Si$_{\text2}$} -\def\e{{\text{\textsc{Elastic}}}} % "elastic" -\def\o{{\text{\textsc{Op}}}} % "order parameter" -\def\i{{\text{\textsc{Int}}}} % "interaction" +\def\e{{\text{\textsc{elastic}}}} % "elastic" +\def\i{{\text{\textsc{int}}}} % "interaction" \def\Dfh{D$_{\text{4h}}$} @@ -40,12 +39,13 @@ \def\op{\textsc{op}} % order parameter \def\ho{\textsc{ho}} % hidden order \def\rus{\textsc{rus}} % Resonant ultrasound spectroscopy +\def\afm{\textsc{afm}} % Antiferromagnetism \def\Rus{\textsc{Rus}} % Resonant ultrasound spectroscopy \def\recip{{\{-1\}}} % functional reciprocal \begin{document} -\title{Elastic properties of \urusi\ are reproduced by modulated $\Bog$ order} +\title{Elastic properties of hidden order in \urusi\ reproduced by modulated $\Bog$ order} \author{Jaron Kent-Dobias} \author{Michael Matty} \author{Brad Ramshaw} @@ -57,101 +57,86 @@ \date\today \begin{abstract} - We develop a phenomenological theory for the elastic response of materials - with a \Dfh\ point group through phase transitions. The physics is - generically that of Lifshitz points, with disordered, uniform ordered, and - modulated ordered phases. Several experimental features of \urusi\ are - reproduced when the order parameter has $\Bog$ symmetry: the topology of the - temperature--pressure phase diagram, the response of the strain stiffness - tensor above the hidden-order transition, and the strain response in the - antiferromagnetic phase. In this scenario, the hidden order is a version of - the high-pressure antiferromagnetic order modulated along the symmetry axis. + We develop a phenomenological mean field theory for the elastic response of + \urusi\ through its hidden order transition. Several experimental features + are reproduced when the order parameter has $\Bog$ symmetry: the topology of + the temperature--pressure phase diagram, the response of the strain stiffness + tensor above the hidden-order transition at zero pressure, and orthorhombic + symmetry breaking in the high-pressure antiferromagnetic phase. In this + scenario, the hidden order is a version of the high-pressure + antiferromagnetic order modulated along the symmetry axis, and the triple + point joining those two phases with the paramagnetic phase is a Lifshitz point. \end{abstract} \maketitle -% \begin{enumerate} -% \item Introduction -% \begin{enumerate} -% \item \urusi\ hidden order intro paragraph, discuss the phase diagram -% \item Strain/OP coupling discussion/RUS -% \item Discussion of experimental data -% \item We look at MFT's for OP's of various symmetries -% \end{enumerate} - -% \item Theory -% \begin{enumerate} -% \item Introduce various pieces of free energy - -% \item Summary of MFT results -% \end{enumerate} - -% \item Data piece - -% \item Talk about more cool stuff like AFM C4 breaking etc -% \end{enumerate} - 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 +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. 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_2019, ikeda_emergent_2012} 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. +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_2019} 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{shekhter_bounding_2013}. -Ref.~\cite{ghosh_single-component_2019} 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 \op. 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{choi_pressure-induced_2018}. Finally, we -discuss our conclusions and the future experimental and theoretical work motivated -by our results. +\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 +\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}. We preform a consistency check for the +applicability of our mean field theory for the \rus\ data. Finally, we discuss +our conclusions and the future experimental and theoretical work motivated by +our results. 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 \op, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is -$f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare stiffness tensor $C$ -tensor is constrained by both that $\epsilon$ is a symmetric tensor and by the -point group symmetry \cite{landau_theory_1995}. The latter can be seen in a -systematic way. First, the six independent components of strain are written as -linear combinations that behave like irreducible representations under the -action of the point group, or +$f_\e=C_{ijkl}\epsilon_{ij}\epsilon_{kl}$, but the form of the bare strain +stiffness tensor $C$ tensor is constrained by both that $\epsilon$ is a +symmetric tensor and by the point group symmetry \cite{landau_theory_1995}. The +latter can be seen in a systematic way. First, the six independent components +of strain are written as linear combinations that behave like irreducible +representations under the action of the point group, or \begin{equation} \begin{aligned} \epsilon_\Aog^{(1)}=\epsilon_{11}+\epsilon_{22} && \hspace{0.1\columnwidth} @@ -187,68 +172,49 @@ representation is $\X$, then the most general coupling to linear order is \end{equation} If $\X$ is a representation not present in the strain there can be no linear coupling, and the effect of $\eta$ going through a continuous phase transition -is to produce a jump in the $\Aog$ strain stiffness. We will therefore focus -our attention on \op\ symmetries that produce linear couplings to -strain. Looking at the components present in \eqref{eq:strain-components}, this rules out all of the u-reps (odd under inversion) and the $\Atg$ irrep as having any anticipatory response in the strain stiffness. +is to produce a jump in the $\Aog$ strain stiffness \cite{luthi_sound_1970, +ramshaw_avoided_2015, shekhter_bounding_2013}. We will therefore focus our +attention on \op\ symmetries that produce linear couplings to strain. Looking +at the components present in \eqref{eq:strain-components}, this rules out all +of the u-reps (odd under inversion) and the $\Atg$ irrep as having any +anticipatory response in the strain stiffness. If the \op\ transforms like $\Aog$, odd terms are allowed in its free energy and any transition will be abrupt and not continuous without -tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quartic +fine-tuning. For $\X$ as any of $\Bog$, $\Btg$, or $\Eg$, the most general quadratic free energy density is \begin{equation} \begin{aligned} - f_\o=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 + f_\op=\frac12\big[&r\eta^2+c_\parallel(\nabla_\parallel\eta)^2 +c_\perp(\nabla_\perp\eta)^2 \\ - &\quad+D_\parallel(\nabla_\parallel^2\eta)^2 - +D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4 + &\qquad\qquad\qquad\quad+D_\perp(\nabla_\perp^2\eta)^2\big]+u\eta^4 \end{aligned} \label{eq:fo} \end{equation} where $\nabla_\parallel=\{\partial_1,\partial_2\}$ transforms like $\Eu$ and -$\nabla_\perp=\partial_3$ transforms like $\Atu$. We'll take $D_\parallel=0$ -since this does not affect the physics at hand. The full free energy functional of $\eta$ and $\epsilon$ is then +$\nabla_\perp=\partial_3$ transforms like $\Atu$. Other quartic terms are +allowed---especially many for an $\Eg$ \op---but we have included only those +terms necessary for stability when either $r$ or $c_\perp$ become negative. The +full free energy functional of $\eta$ and $\epsilon$ is then \begin{equation} \begin{aligned} F[\eta,\epsilon] - &=F_\o[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\ - &=\int dx\,(f_\o+f_\e+f_\i) + &=F_\op[\eta]+F_\e[\epsilon]+F_\i[\eta,\epsilon] \\ + &=\int dx\,(f_\op+f_\e+f_\i) \end{aligned} \end{equation} -Neglecting interaction terms -higher than quadratic order, the only strain relevant to the problem is -$\epsilon_\X$, and this can be traced out of the problem exactly, since +The only strain relevant to the \op\ is $\epsilon_\X$, which can be traced out +of the problem exactly in mean field theory. Extremizing with respect to +$\epsilon_\X$, \begin{equation} - 0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}=C_\X\epsilon_{\X i}(x) + 0=\frac{\delta F[\eta,\epsilon]}{\delta\epsilon_{\X i}(x)}\bigg|_{\epsilon=\epsilon_\star}=C_\X\epsilon^\star_{\X i}(x) +\frac12b\eta_i(x) \end{equation} -gives $\epsilon_\X[\eta]=-(b/2C_\X)\eta$. Upon substitution into the free -energy, the resulting effective free energy $F[\eta,\epsilon[\eta]]$ has a density identical to $f_\o$ -with $r\to\tilde r=r-b^2/4C_\X$. - -With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz -point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, -lifshitz_theory_1942-1}. For a one-component \op\ ($\Bog$ or $\Btg$) it is -traditional to make the field ansatz $\eta(x)=\eta_*\cos(q_*x_3)$. For $\tilde -r>0$ and $c_\perp>0$, or $\tilde r<c_\perp^2/4D_\perp$ and $c_\perp<0$, the -only stable solution is $\eta_*=q_*=0$ and the system is unordered. For $\tilde -r<0$ there are free energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and -this system has uniform order. For $c_\perp<0$ and $\tilde -r<c_\perp^2/4D_\perp$ there are free energy minima for -$q_*^2=-c_\perp/2D_\perp$ and -\begin{equation} - \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u} - =\frac{\tilde r_c-\tilde r}{3u} - =\frac{\Delta\tilde r}{3u} -\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 -field 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. In this case the uniform ordered phase is only -stable for $c_\perp>0$, and the modulated phase is now characterized by helical -order with $\eta(x)=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. -The uniform--modulated transition is now continuous. This already does not reproduce the physics of \ho, and so we will henceforth neglect this possibility. The schematic phase -diagrams for this model are shown in Figure \ref{fig:phases}. +gives the optimized strain conditional on the \op\ as +$\epsilon_\X^\star[\eta](x)=-(b/2C_\X)\eta(x)$ and $\epsilon_\Y^\star[\eta]=0$ +for all other $\Y$. Upon substitution into the free energy, the resulting +effective free energy $F[\eta,\epsilon_\star[\eta]]$ has a density identical to +$f_\op$ with $r\to\tilde r=r-b^2/4C_\X$. \begin{figure}[htpb] \includegraphics[width=\columnwidth]{phase_diagram_experiments} @@ -270,28 +236,62 @@ diagrams for this model are shown in Figure \ref{fig:phases}. \label{fig:phases} \end{figure} -We will now proceed to derive the effective strain stiffness tensor $\lambda$ that results from the coupling of strain to the \op. The ultimate result, in \eqref{eq:elastic.susceptibility}, is that $\lambda_\X$ only differs from its bare value for the symmetry $\X$ of the order parameter. 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 strain stiffness using some tricks from functional calculus. - -The susceptibility of the order parameter to a field linearly coupled to it is given by +With the strain traced out \eqref{eq:fo} describes the theory of a Lifshitz +point at $\tilde r=c_\perp=0$ \cite{lifshitz_theory_1942, +lifshitz_theory_1942-1}. For a one-component \op\ ($\Bog$ or $\Btg$) it is +traditional to make the field ansatz +$\langle\eta(x)\rangle=\eta_*\cos(q_*x_3)$. For $\tilde r>0$ and $c_\perp>0$, +or $\tilde r>c_\perp^2/4D_\perp$ and $c_\perp<0$, the only stable solution is +$\eta_*=q_*=0$ and the system is unordered. For $\tilde r<0$ there are free +energy minima for $q_*=0$ and $\eta_*^2=-\tilde r/4u$ and this system has +uniform order. For $c_\perp<0$ and $\tilde r<c_\perp^2/4D_\perp$ there are free +energy minima for $q_*^2=-c_\perp/2D_\perp$ and +\begin{equation} + \eta_*^2=\frac{c_\perp^2-4D_\perp\tilde r}{12D_\perp u} + =\frac{\tilde r_c-\tilde r}{3u} + =\frac{|\Delta\tilde r|}{3u} +\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 field 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. In this case the uniform ordered phase +is only stable for $c_\perp>0$, and the modulated phase is now characterized by +helical order with $\langle\eta(x)\rangle=\eta_*\{\cos(q_*x_3),\sin(q_*x_3)\}$. +The uniform--modulated transition is now continuous. This does not +reproduce the physics of \ho, which has an abrupt transition between \ho\ and \afm, and so we will henceforth neglect the possibility of a multicomponent order parameter. The schematic phase diagrams for this model are shown in Figure +\ref{fig:phases}. + +We will now proceed to derive the \emph{effective strain stiffness 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. 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 +strain stiffness using some tricks from functional calculus. + +The susceptibility of a single component ($\Bog$ or $\Btg$) \op\ to a field +linearly coupled to it is given by \begin{equation} \begin{aligned} &\chi^\recip(x,x') - =\frac{\delta^2F[\eta,\epsilon[\eta]]}{\delta\eta(x)\delta\eta(x')} - =\big(\tilde r-c_\parallel\nabla_\parallel^2-c_\perp\nabla_\perp^2 \\ - &\qquad\qquad+D_\perp\nabla_\perp^4+12u\eta^2(x)\big) + =\frac{\delta^2F[\eta,\epsilon_\star[\eta]]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\langle\eta\rangle} + =\big[\tilde r-c_\parallel\nabla_\parallel^2 \\ + &\qquad\qquad-c_\perp\nabla_\perp^2+D_\perp\nabla_\perp^4+12u\langle\eta(x)\rangle^2\big] \delta(x-x'), \end{aligned} \label{eq:sus_def} \end{equation} where $\recip$ indicates a \emph{functional reciprocal} in the sense that -\[ +\begin{equation} \int dx''\,\chi^\recip(x,x'')\chi(x'',x')=\delta(x-x'). -\] +\end{equation} Taking the Fourier transform and integrating over $q'$ we have \begin{equation} \chi(q) =\big(\tilde r+c_\parallel q_\parallel^2+c_\perp q_\perp^2+D_\perp q_\perp^4 - +12u\sum_{q'}\tilde\eta_{q'}\tilde\eta_{-q'}\big)^{-1}. + +12u\sum_{q'}\langle\tilde\eta_{q'}\rangle\langle\tilde\eta_{-q'}\rangle\big)^{-1}. \end{equation} Near the unordered--modulated transition this yields \begin{equation} @@ -305,89 +305,111 @@ Near the unordered--modulated transition this yields \label{eq:susceptibility} \end{equation} with $\xi_\perp=(|\Delta\tilde r|/D_\perp)^{-1/4}$ and -$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. We must emphasize that this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\ cannot couple linearly to a uniform magnetic field. The object defined in \eqref{eq:sus_def} is most readily interpreted as proportional to the two-point connected correlation function $\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$. +$\xi_\parallel=(|\Delta\tilde r|/c_\parallel)^{-1/2}$. We must emphasize that +this is \emph{not} the magnetic susceptibility because a $\Bog$ or $\Btg$ \op\ +cannot couple linearly to a uniform magnetic field. The object defined in +\eqref{eq:sus_def} is most readily interpreted as proportional to the two-point +connected correlation function +$\langle\delta\eta(x)\delta\eta(x')\rangle=G(x,x')=k_BT\chi(x,x')$. The strain stiffness is given in a similar way to the inverse susceptibility: we must trace over $\eta$ and take the second variation of the resulting free energy functional of $\epsilon$. Extremizing over $\eta$ yields \begin{equation} - 0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}=\frac{\delta F_\o[\eta]}{\delta\eta(x)} - +\frac12b\epsilon_\X(x), + 0=\frac{\delta F[\eta,\epsilon]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}= + \frac12b\epsilon_\X(x)+\frac{\delta F_\op[\eta]}{\delta\eta(x)}\bigg|_{\eta=\eta_\star}, \label{eq:implicit.eta} \end{equation} -which implicitly gives $\eta[\epsilon]$. Since $\eta$ is a functional of $\epsilon_\X$ alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. Though this -cannot be solved explicitly, we can make use of the inverse function theorem. -First, denote by $\eta^{-1}[\eta]$ the inverse functional of $\eta$ implied by +which implicitly gives $\eta_\star[\epsilon]$, the optimized \op\ conditioned on the strain. Since $\eta_\star$ is a functional of $\epsilon_\X$ +alone, only the stiffness $\lambda_\X$ is modified from its bare value $C_\X$. +Though this differential equation for $\eta_*$ cannot be solved explicitly, we +can make use of the inverse function theorem. First, denote by +$\eta_\star^{-1}[\eta]$ the inverse functional of $\eta_\star$ implied by \eqref{eq:implicit.eta}, which gives the function $\epsilon_\X$ corresponding -to each solution of \eqref{eq:implicit.eta} it receives. This we can immediately identify from \eqref{eq:implicit.eta} as $\eta^{-1}[\eta](x)=-2/b(\delta F_\o[\eta]/\delta\eta(x))$. -Now, we use the inverse function -theorem to relate the functional reciprocal of the derivative of $\eta[\epsilon]$ with respect to $\epsilon_\X$ to the derivative of $\eta^{-1}[\eta]$ with respect to $\eta$, yielding +to each solution of \eqref{eq:implicit.eta} it receives. This we can +immediately identify from \eqref{eq:implicit.eta} as +$\eta^{-1}_\star[\eta](x)=-2/b(\delta F_\op[\eta]/\delta\eta(x))$. Now, we use +the inverse function theorem to relate the functional reciprocal of the +derivative of $\eta_\star[\epsilon]$ with respect to $\epsilon_\X$ to the +derivative of $\eta^{-1}_\star[\eta]$ with respect to $\eta$, yielding \begin{equation} \begin{aligned} - \bigg(\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}\bigg)^\recip - &=\frac{\delta\eta^{-1}[\eta](x)}{\delta\eta(x')} - =-\frac2b\frac{\delta^2F_\o[\eta]}{\delta\eta(x)\delta\eta(x')} \\ - &=-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'), + \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip + &=\frac{\delta\eta_\star^{-1}[\eta](x)}{\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]} + =-\frac2b\frac{\delta^2F_\op[\eta]}{\delta\eta(x)\delta\eta(x')}\bigg|_{\eta=\eta^*[\epsilon]}. \end{aligned} \label{eq:inv.func} \end{equation} -where we have used what we already know about the variation of $F_\o[\eta]$ with respect to $\eta$. -Finally, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} can be used in concert with the ordinary rules of functional calculus to yield the strain stiffness +Next, \eqref{eq:implicit.eta} and \eqref{eq:inv.func} +can be used in concert with the ordinary rules of functional calculus to yield +the second variation \begin{widetext} \begin{equation} \begin{aligned} - \lambda_\X(x,x') - &=\frac{\delta^2F[\eta[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ + \frac{\delta^2F[\eta_\star[\epsilon],\epsilon]}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} &=C_\X\delta(x-x')+ - b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')} - +\frac12b\int dx''\,\epsilon_{\X k}(x'')\frac{\delta^2\eta_k(x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')} \\ - &\qquad+\int dx''\,dx'''\,\frac{\delta^2F_\o[\eta]}{\delta\eta(x'')\delta\eta(x''')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} - +\int dx''\,\frac{\delta F_\o[\eta]}{\delta\eta(x'')}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')} \\ + b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} + +\frac12b\int dx''\,\frac{\delta^2\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')\delta\epsilon_\X(x'')}\epsilon_\X(x'') \\ + &\quad+\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\frac{\delta^2F_\op[\eta]}{\delta\eta(x'')\delta\eta(x''')}\bigg|_{\eta=\eta_\star[\epsilon]} + +\int dx''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)\delta\epsilon_\X(x')}\frac{\delta F_\op[\eta]}{\delta\eta(x'')}\bigg|_{\eta=\eta_\star[\epsilon]} \\ &=C_\X\delta(x-x')+ - b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')} - -\frac12b\int dx''\,dx'''\,\bigg(\frac{\partial\eta(x'')}{\partial\epsilon_\X(x''')}\bigg)^{-1}\frac{\delta\eta(x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta(x''')}{\delta\epsilon_\X(x')} \\ + b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} + -\frac12b\int dx''\,dx'''\,\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x)}\frac{\delta\eta_\star[\epsilon](x''')}{\delta\epsilon_\X(x')}\bigg(\frac{\partial\eta_\star[\epsilon](x'')}{\partial\epsilon_\X(x''')}\bigg)^\recip \\ &=C_\X\delta(x-x')+ - b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')} - -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta(x'')}{\delta\epsilon_\X(x')} + b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')} + -\frac12b\int dx''\,\delta(x-x'')\frac{\delta\eta_\star[\epsilon](x'')}{\delta\epsilon_\X(x')} =C_\X\delta(x-x')+ - \frac12b\frac{\delta\eta(x)}{\delta\epsilon_\X(x')}, + \frac12b\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}. \end{aligned} + \label{eq:big.boy} \end{equation} \end{widetext} -whose Fourier transform follows from \eqref{eq:inv.func} as +The strain stiffness is given by the second variation evaluated at the +extremized solution $\langle\epsilon\rangle$. To calculate it, note that +evaluating the second variation of $F_\op$ in \eqref{eq:inv.func} at +$\langle\epsilon\rangle$ (or +$\eta_\star(\langle\epsilon\rangle)=\langle\eta\rangle$) yields +\begin{equation} + \bigg(\frac{\delta\eta_\star[\epsilon](x)}{\delta\epsilon_\X(x')}\bigg)^\recip\bigg|_{\epsilon=\langle\epsilon\rangle} + =-\frac2b\chi^\recip(x,x')-\frac{b}{2C_\X}\delta(x-x'). + \label{eq:recip.deriv.op} +\end{equation} +Upon substitution into \eqref{eq:big.boy} and taking the Fourier transform of +the result, we finally arrive at \begin{equation} - \lambda_\X(q)=C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}. + \lambda_\X(q) + =C_\X-\frac b2\bigg(\frac2{b\chi(q)}+\frac b{2C_\X}\bigg)^{-1} + =C_\X\bigg(1+\frac{b^2}{4C_\X}\chi(q)\bigg)^{-1}. \label{eq:elastic.susceptibility} \end{equation} -Though not relevant here, this result generalizes to multicomponent order parameters. -At $q=0$, which is where the stiffness measurements used here were taken, this -predicts a cusp in the elastic susceptibility of the form $|\Delta\tilde r|^\gamma$ for $\gamma=1$. +Though not relevant here, this result generalizes to multicomponent \op s. At +$q=0$, which is where the stiffness measurements used here were taken, this +predicts a cusp in the strain stiffness of the form $|\Delta\tilde +r|^\gamma$ for $\gamma=1$. \begin{figure}[htpb] \centering \includegraphics[width=\columnwidth]{fig-stiffnesses} \caption{ Measurements of the effective strain stiffness as a function of temperature - for the six independent components of strain from \rus. The vertical - lines show the location of the \ho\ transition. + for the six independent components of strain from \rus. The vertical lines + show the location of the \ho\ transition. } \label{fig:data} \end{figure} -We have seen that mean field theory predicts that whatever component of strain -transforms like the \op\ will see a $t^{-1}$ softening in the -stiffness that ends in a cusp. \Rus\ experiments \cite{ghosh_single-component_2019} -yield the strain stiffness for various components of the strain; this data is -shown in Figure \ref{fig:data}. The $\Btg$ and $\Eg$ stiffnesses don't appear -to have any response to the presence of the transition, exhibiting the expected -linear stiffening with a low-temperature cutoff \cite{varshni_temperature_1970}. The $\Bog$ stiffness has a dramatic response, softening over the -course of roughly $100\,\K$. There is a kink in the curve right at the -transition. While the low-temperature response is not as dramatic as the theory -predicts, mean field theory---which is based on a small-$\eta$ expansion---will -not work quantitatively far below the transition where $\eta$ has a large -nonzero value and higher powers in the free energy become important. The data -in the high-temperature phase can be fit to the theory -\eqref{eq:elastic.susceptibility}, with a linear background stiffness -$C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is +\Rus\ experiments \cite{ghosh_single-component_nodate} yield the strain +stiffness for various components of the strain; this data is shown in Figure +\ref{fig:data}. The $\Btg$ stiffness doesn't appear to have any response to +the presence of the transition, exhibiting the expected linear stiffening with +a low-temperature cutoff \cite{varshni_temperature_1970}. The $\Bog$ stiffness +has a dramatic response, softening over the course of roughly $100\,\K$. There +is a kink in the curve right at the transition. While the low-temperature +response is not as dramatic as the theory predicts, mean field theory---which +is based on a small-$\eta$ expansion---will not work quantitatively far below +the transition where $\eta$ has a large nonzero value and higher powers in the +free energy become important. The data in the high-temperature phase can be fit +to the theory \eqref{eq:elastic.susceptibility}, with a linear background +stiffness $C_\Bog^{(11)}$ and $\tilde r-\tilde r_c=a(T-T_c)$, and the result is shown in Figure \ref{fig:fit}. The data and theory appear consistent. \begin{figure}[htpb] @@ -398,29 +420,75 @@ shown in Figure \ref{fig:fit}. The data and theory appear consistent. (dashed). The fit gives $C_\Bog^{(11)}\simeq\big[71-(0.010\,\K^{-1})T\big]\,\GPa$, $b^2/4D_\perp q_*^4\simeq6.2\,\GPa$, and $a/D_\perp - q_*^4\simeq0.0038\,\K^{-1}$. + q_*^4\simeq0.0038\,\K^{-1}$. The failure of the Ginzburg--Landau prediction + below the transition is expected on the grounds that the \op\ is too large + for the free energy expansion to be valid by the time the Ginzburg + temperature is reached. } \label{fig:fit} \end{figure} -Mean field theory neglects the effect of fluctuations on critical behavior, yet -also predicts the magnitude of those fluctuations. This allows a mean field -theory to undergo an internal consistency check to ensure the predicted -fluctuations are indeed negligible. This is typically done by computing the -Ginzburg criterion \cite{ginzburg_remarks_1961}, which gives the proximity to -the critical point $t=(T-T_c)/T_c$ at which mean field theory is expected to -break down by comparing the magnitude of fluctuations in a correlation-length -sized box to the magnitude of the field. In the modulated phase the spatially -averaged magnitude is zero, and so we will instead compare fluctuations in the +We have seen that the mean-field theory of a $\Bog$ \op\ recreates the topology +of the \ho\ phase diagram and the temperature dependence of the $\Bog$ strain +stiffness at zero pressure. There are several implications of this theory. First, +the association of a modulated $\Bog$ order with the \ho\ phase implies a +\emph{uniform} $\Bog$ order associated with the \afm\ phase, and moreover a +uniform $\Bog$ strain of magnitude $\langle\epsilon_\Bog\rangle^2=b^2\tilde +r/16uC_\Bog^2$, which corresponds to an orthorhombic phase. Orthorhombic +symmetry breaking was recently detected in the \afm\ phase of \urusi\ using +x-ray diffraction, a further consistency of this theory with the phenomenology +of \urusi\ \cite{choi_pressure-induced_2018}. Second, as the Lifshitz point is +approached from low pressure this theory predicts the modulation wavevector +$q_*$ should continuously vanish. Far 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, or $q_*\simeq0.328\,\A^{-1}$ \cite{meng_imaging_2013}. In between +these two regimes, the ordering wavevector should shrink by jumping between +ever-closer commensurate values in the style of the devil's staircase +\cite{bak_commensurate_1982}. This motivates future \rus\ experiments done at +pressure, where the depth of the cusp in the $\Bog$ stiffness should deepen +(perhaps with these commensurability jumps) at low pressure and approach zero +like $q_*^4\sim(c_\perp/2D_\perp)^2$ near the Lifshitz point. The presence of +spatial commensurability is not expected to modify the critical behavior otherwise +\cite{garel_commensurability_1976}. + +There are two apparent discrepancies between the orthorhombic strain in the +phase diagram presented by \cite{choi_pressure-induced_2018} and that predicted +by our mean field theory. The first is the apparent onset of the orthorhombic +phase in the \ho\ state prior to the onset of \afm. As +\cite{choi_pressure-induced_2018} notes, this could be due to the lack of +an ambient pressure calibration for the lattice constant. The second +discrepancy is the onset of orthorhombicity at higher temperatures than the +onset of \afm. Susceptibility data sees no trace of another phase transition at +these higher temperatures \cite{inoue_high-field_2001}, and therefore we don't +in fact expect there to be one. We do expect that this could be due to the high +energy nature of x-rays as an experimental probe: orthorhombic fluctuations +could appear at higher temperatures than the true onset of an orthorhombic +phase. + +Three dimensions is below the upper critical dimension $4\frac12$, and so mean +field theory should break down sufficiently close to the critical point due to +fluctuations \cite{hornreich_lifshitz_1980}. Mean field theory neglects the +effect of fluctuations on critical behavior, yet also predicts the magnitude of +those fluctuations. This allows a mean field theory to undergo an internal +consistency check to ensure the predicted fluctuations are indeed negligible in the region under consideration. +This is typically done by computing the Ginzburg temperature +\cite{ginzburg_remarks_1961}, which gives the proximity to the critical point +$t=(T-T_c)/T_c$ at which mean field theory is expected to break down by +comparing the magnitude of fluctuations in a correlation-length sized box to +the magnitude of the field. In the modulated phase the spatially averaged +magnitude is zero, and so we will instead compare fluctuations in the \emph{amplitude} at $q_*$ to the magnitude of that amplitude. Defining the field $\alpha$ by $\eta(x)=\alpha(x)e^{-iq_*x_3}$, it follows that in the -modulated phase $\alpha(x)=\alpha_0$ for $\alpha_0^2=|\delta \tilde r|/4u$. In -the modulated phase, the $q$-dependant fluctuations in $\alpha$ are given by -\[ +modulated phase $\langle\alpha(x)\rangle=\alpha_0$ for $\alpha_0^2=|\delta +\tilde r|/4u$. In the modulated phase, the $q$-dependant fluctuations in +$\alpha$ are given by +\begin{equation} G_\alpha(q)=k_BT\chi_\alpha(q)=\frac1{c_\parallel q_\parallel^2+D_\perp(4q_*^2q_\perp^2+q_\perp^4)+2|\delta r|}, -\] +\end{equation} An estimate of the Ginzburg criterion is then given by the temperature at which -$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle\simeq\langle\alpha\rangle^2=\alpha_0^2$, +$V_\xi^{-1}\int_{V_\xi}G_\alpha(0,x)\,dx=\langle\delta\alpha^2\rangle_{V_\xi}\simeq\langle\alpha\rangle^2=\alpha_0^2$, where $V_\xi=\xi_\perp\xi_\parallel^2$ is a correlation volume. The parameter $u$ can be replaced in favor of the jump in the specific heat at the transition using @@ -428,37 +496,57 @@ using c_V=-T\frac{\partial^2f}{\partial T^2} =\begin{cases}0&T>T_c\\Ta^2/12 u&T<T_c.\end{cases} \end{equation} -The integral over the correlation function $G_\alpha$ can be preformed up to one integral analytically using a Gaussian-bounded correlation volume, yielding $\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}\delta r^{-1}\mathcal I(\xi_\perp q_*)$ for -\[ - \mathcal I(x)=-2^{3/2}\pi^{5/2}\int dy\,e^{[2+(4x^2-1)y^2+y^4]/2} -\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big) -\] +The integral over the correlation function $G_\alpha$ can be preformed up to +one integral analytically using a Gaussian-bounded correlation volume, yielding +$\langle\Delta\tilde r\rangle^2\simeq k_BTV_\xi^{-1}|\Delta \tilde r|^{-1}\mathcal +I(\xi_\perp q_*)$ for +\begin{equation} + \begin{aligned} + \mathcal I(x)=-2^{3/2}\pi^{5/2}\int& dy\,e^{[2+(4x^2-1)y^2+y^4]/2} \\ + &\times\mathop{\mathrm{Ei}}\big(-(1+2x^2y^2+\tfrac12y^4)\big). + \end{aligned} +\end{equation} This gives a transcendental equation -\[ - \mathcal I(\xi_{\perp0}q_*\delta t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}\delta t_\G^{3/4}. -\] +\begin{equation} + \mathcal I(\xi_{\perp0}q_*t_\G)\simeq3k_B^{-1}\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}t_\G^{3/4}, +\end{equation} +with $\xi=\xi_0|t|^{-\nu}$ defining the bare correlation lengths. Experiments give $\Delta c_V\simeq1\times10^5\,\J\,\m^{-3}\,\K^{-1}$ -\cite{fisher_specific_1990}, and our fit above gives $\xi_{\perp0}q_*=(D_\perp -q_*^4/aT_c)^{1/4}\simeq2$. We have reason to believe that at zero pressure, very -far from the Lifshitz point, the half-wavelength of the modulation should be commensurate with the lattice, giving $q_*\simeq0.328\,\A^{-1}$ -\cite{meng_imaging_2013}. Further supposing that $\xi_{\parallel0}\simeq\xi_{\perp0}$, -we find $\delta t_\G\sim0.4$, though this estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in $\delta t_\G$ over nearly four orders of magnitude. -The estimate here predicts that an experiment may begin to see deviations from -mean field behavior within around $5\,\K$ of the critical point. A \rus\ -experiment with more precise temperature resolution near the critical point may -be able to resolve a modified cusp exponent $\gamma\simeq1.31$ \cite{guida_critical_1998}, since the universality class of a uniaxial modulated scalar \op\ is $\mathrm O(2)$ \cite{garel_commensurability_1976}. Our work here appears self--consistent, given that our fit is mostly concerned with temperatures farther than this from the critical point. This analysis also indicates that 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. - -There are two apparent discrepancies between the phase diagram presented in -\cite{choi_pressure-induced_2018} and that predicted by our mean field theory. The first is the apparent -onset of the orthorhombic phase in the \ho\ state prior to the onset of AFM. -As ref.\cite{choi_pressure-induced_2018} notes, this could be due to the lack of an ambient pressure calibration -for the lattice constant. The second discrepancy is the onset of orthorhombicity -at higher temperatures than the onset of AFM. Susceptibility data sees no trace of another phase transition at these higher temperatures \cite{inoue_high-field_2001}, and therefore we don't in fact expect there to be one. We do expect that this could be due to the -high energy nature of x-rays as an experimental probe: orthorhombic fluctuations -could appear at higher temperatures than the true onset of an orthorhombic phase. +\cite{fisher_specific_1990}. Our fit above gives $\xi_{\perp0}q_*=(D_\perp +q_*^4/aT_c)^{1/4}\simeq2$. Further supposing that +$\xi_{\parallel0}\simeq\xi_{\perp0}$, we find $t_\G\sim0.4$, though this +estimate is sensitive to uncertainty in $\xi_{\parallel0}$: varying our +estimate for $\xi_{\parallel0}$ over one order of magnitude yields changes in +$t_\G$ over nearly four orders of magnitude. The estimate here predicts +that an experiment may begin to see deviations from mean field behavior within +around several degrees Kelvin of the critical point. A \rus\ experiment with more precise +temperature resolution near the critical point may be able to resolve a +modified cusp exponent $\gamma\simeq1.31$ \cite{guida_critical_1998}, since the +universality class of a uniaxial modulated scalar \op\ is $\mathrm O(2)$ +\cite{garel_commensurability_1976}. Our work here appears self--consistent, +given that our fit is mostly concerned with temperatures ten to hundreds of +Kelvin from the critical point. This analysis also indicates that 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. + +We have preformed a general treatment of phenomenological \ho\ \op s with the +potential for linear coupling to strain. The possibilities with consistent mean +field phase diagrams are $\Bog$ and $\Btg$, and the only of these consistent +with zero-pressure \rus\ data is $\Bog$, with a cusp appearing in the +associated stiffness. In this picture, the \ho\ phase is characterized by +uniaxial modulated $\Bog$ order, while the \afm\ phase is characterized by +uniform $\Bog$ order. The corresponding prediction of uniform $\Bog$ symmetry +breaking in the \afm\ phase is consistent with recent diffraction experiments +\cite{choi_pressure-induced_2018}. This work motivates both further theoretical +work regarding a microscopic theory with modulated $\Bog$ order, and preforming +\rus\ experiments at pressure that could further support or falsify this idea. \begin{acknowledgements} - + This research was supported by NSF DMR-1719490, [Mike's grant], [Brad's + grants????]. The authors would like to thank [ask Brad] for helpful + correspondence. \end{acknowledgements} \bibliographystyle{apsrev4-1} |