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| author | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2019-08-05 21:47:14 -0400 |
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| committer | Jaron Kent-Dobias <jaron@kent-dobias.com> | 2019-08-05 21:47:14 -0400 |
| commit | 64ebc5efdb0498c459cc1d280a9acac62b68f151 (patch) | |
| tree | e4023852710add5f48ae5d9f20b37a77f08eb1e7 | |
| parent | 701fe2f91798abbddd1082e21c760f69afc9ad31 (diff) | |
| download | PRB_102_075129-64ebc5efdb0498c459cc1d280a9acac62b68f151.tar.gz PRB_102_075129-64ebc5efdb0498c459cc1d280a9acac62b68f151.tar.bz2 PRB_102_075129-64ebc5efdb0498c459cc1d280a9acac62b68f151.zip | |
redo of the phase diagrams
| -rw-r--r-- | hidden_order.bib | 332 | ||||
| -rw-r--r-- | main.tex | 163 | ||||
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| -rw-r--r-- | phases_scalar.pdf | bin | 10422 -> 11640 bytes | |||
| -rw-r--r-- | phases_vector.pdf | bin | 9666 -> 9768 bytes | |||
| -rw-r--r-- | plots.nb | 1609 |
6 files changed, 2026 insertions, 78 deletions
diff --git a/hidden_order.bib b/hidden_order.bib index 3057a7c..436345a 100644 --- a/hidden_order.bib +++ b/hidden_order.bib @@ -92,4 +92,336 @@ file = {/home/pants/.zotero/data/storage/JVMTIZGB/Ginzburg - 1961 - Some Remarks on Phase Transitions of the Second Ki.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.}, + 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{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 = {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{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/data/storage/WV2NUDE9/Ghosh et al. - 2019 - Single-Component Order Parameter in URu$_2$Si$_2$ .pdf} +} + +@article{ikeda_emergent_2012, + title = {Emergent Rank-5 Nematic Order in {{URu}}{\textsubscript{2}}{{Si}}{\textsubscript{2}}}, + volume = {8}, + issn = {1745-2481}, + abstract = {Exotic electronic states resulting from entangled spin and orbital degrees of freedom are hallmarks of strongly correlated f-electron systems. A spectacular example is the so-called hidden-order (HO) phase transition1 in the heavy-electron metal URu2Si2, which is characterized by the huge amount of entropy lost at THO=17.5 K (refs 2, 3). However, no evidence of magnetic/structural phase transition has been found below THO so far. The origin of the HO phase transition has been a long-standing mystery in condensed-matter physics. Here, on the basis of a first-principles theoretical approach, we examine the complete set of multipole correlations allowed in this material. The results uncover that the HO parameter is a rank-5 multipole (dotriacontapole) order with nematic E- symmetry, which exhibits staggered pseudospin moments along the [110] direction. This naturally provides comprehensive explanations of all key features in the HO phase including anisotropic magnetic excitations, the nearly degenerate antiferromagnetic-ordered state and spontaneous rotational-symmetry breaking.}, + language = {en}, + number = {7}, + journal = {Nature Physics}, + doi = {10/f34f8m}, + author = {Ikeda, Hiroaki and Suzuki, Michi-To and Arita, Ryotaro and Takimoto, Tetsuya and Shibauchi, Takasada and Matsuda, Yuji}, + month = jul, + year = {2012}, + pages = {528-533}, + file = {/home/pants/.zotero/data/storage/9NYNGB45/Ikeda et al. - 2012 - Emergent rank-5 nematic order in URusub2subSi.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/data/storage/39JAA4F8/Harrison and Jaime - 2019 - Hidden valence transition in URu2Si2.pdf} +} + +@article{chandra_hastatic_2013, + title = {Hastatic Order in the Heavy-Fermion Compound {{URu}}{\textsubscript{2}}{{Si}}{\textsubscript{2}}}, + volume = {493}, + issn = {1476-4687}, + abstract = {The development of collective long-range order by means of phase transitions occurs by the spontaneous breaking of fundamental symmetries. Magnetism is a consequence of broken time-reversal symmetry, whereas superfluidity results from broken gauge invariance. The broken symmetry that develops below 17.5 kelvin in the heavy-fermion compound URu2Si2 has long eluded such identification. Here we show that the recent observation of Ising quasiparticles in URu2Si2 results from a spinor order parameter that breaks double time-reversal symmetry, mixing states of integer and half-integer spin. Such `hastatic' order hybridizes uranium-atom conduction electrons with Ising 5f2 states to produce Ising quasiparticles; it accounts for the large entropy of condensation and the magnetic anomaly observed in torque magnetometry. Hastatic order predicts a tiny transverse moment in the conduction-electron `sea', a colossal Ising anisotropy in the nonlinear susceptibility anomaly and a resonant, energy-dependent nematicity in the tunnelling density of states.}, + language = {en}, + number = {7434}, + journal = {Nature}, + doi = {10/gf5vbj}, + author = {Chandra, Premala and Coleman, Piers and Flint, Rebecca}, + month = jan, + year = {2013}, + pages = {621-626}, + file = {/home/pants/.zotero/data/storage/B272KFL9/Chandra et al. - 2013 - Hastatic order in the heavy-fermion compound URus.pdf} +} + +@article{ikeda_theory_1998, + title = {Theory of {{Unconventional Spin Density Wave}}: {{A Possible Mechanism}} of the {{Micromagnetism}} in {{U}}-Based {{Heavy Fermion Compounds}}}, + volume = {81}, + shorttitle = {Theory of {{Unconventional Spin Density Wave}}}, + abstract = {We propose a novel spin density wave (SDW) state as a possible mechanism of the anomalous antiferromagnetism, the so called micromagnetism, in URu2Si2 below 17.5 K. In this new SDW, the electron-hole pair amplitude changes its sign in the momentum space as in the case of the unconventional superconductivity. It is shown that this state can be realized in an extended Hubbard model within the mean field theory. We also examine some characteristic properties of this SDW to compare with the experimental results. All these properties well explain the unsolved problem of the micromagnetism.}, + number = {17}, + journal = {Physical Review Letters}, + doi = {10/bw6hn5}, + author = {Ikeda, Hiroaki and Ohashi, Yoji}, + month = oct, + year = {1998}, + pages = {3723-3726}, + file = {/home/pants/.zotero/data/storage/QNE8NK4Q/Ikeda and Ohashi - 1998 - Theory of Unconventional Spin Density Wave A Poss.pdf} +} + +@article{hoshino_resolution_2013, + title = {Resolution of {{Entropy}} \textbackslash{}(\textbackslash{}ln\textbackslash{}sqrt\{2\}\textbackslash{}) by {{Ordering}} in {{Two}}-{{Channel Kondo Lattice}}}, + volume = {82}, + issn = {0031-9015}, + abstract = {Peculiar property of electronic order is clarified for the two-channel Kondo lattice. With two conduction electrons per site, the order parameter is a composite quantity involving both local and itinerant degrees of freedom. In contrast to the ordinary Kondo lattice, a heavy electron band is absent above the transition temperature, but is rapidly formed below it. The change of entropy associated with the ordering is found to be close to ln2\textendash{$\surd$}ln2\textbackslash{}ln \textbackslash{}sqrt\{2\} per site. This entropy corresponds to the residual entropy in a two-channel Kondo impurity, which has been regarded as due to localized free Majorana particles. The present composite order is interpreted as instability of Majorana particles toward non-Kramers conduction electrons plus heavy fermions that involve localized electrons.}, + number = {4}, + journal = {Journal of the Physical Society of Japan}, + doi = {10/gf5vbk}, + author = {Hoshino, Shintaro and Otsuki, Junya and Kuramoto, Yoshio}, + month = mar, + year = {2013}, + pages = {044707}, + file = {/home/pants/.zotero/data/storage/TY637XGC/Hoshino et al. - 2013 - Resolution of Entropy (lnsqrt 2 ) by Ordering .pdf} +} + +@article{riggs_evidence_2015, + title = {Evidence for a Nematic Component to the Hidden-Order Parameter in {{URu}}{\textsubscript{2}}{{Si}}{\textsubscript{2}} from Differential Elastoresistance Measurements}, + volume = {6}, + issn = {2041-1723}, + abstract = {For materials that harbour a continuous phase transition, the susceptibility of the material to various fields can be used to understand the nature of the fluctuating order and hence the nature of the ordered state. Here we use anisotropic biaxial strain to probe the nematic susceptibility of URu2Si2, a heavy fermion material for which the nature of the low temperature `hidden order' state has defied comprehensive understanding for over 30 years. Our measurements reveal that the fluctuating order has a nematic component, confirming reports of twofold anisotropy in the broken symmetry state and strongly constraining theoretical models of the hidden-order phase.}, + language = {en}, + journal = {Nature Communications}, + doi = {10/gf5vbm}, + author = {Riggs, Scott C. and Shapiro, M. C. and Maharaj, Akash V. and Raghu, S. and Bauer, E. D. and Baumbach, R. E. and {Giraldo-Gallo}, P. and Wartenbe, Mark and Fisher, I. R.}, + month = mar, + year = {2015}, + pages = {6425}, + file = {/home/pants/.zotero/data/storage/Z57IE8J9/Riggs et al. - 2015 - Evidence for a nematic component to the hidden-ord.pdf} +} + +@article{rau_hidden_2012, + title = {Hidden and Antiferromagnetic Order as a Rank-5 Superspin in {{URu}}\$\{\}\_\{2\}\${{Si}}\$\{\}\_\{2\}\$}, + volume = {85}, + abstract = {We propose a candidate for the hidden order in URu2Si2: a rank-5 E type spin-density wave between uranium 5f crystal-field doublets {$\Gamma$}(1)7 and {$\Gamma$}(2)7, breaking time-reversal and lattice tetragonal symmetry in a manner consistent with recent torque measurements [Okazaki et al., Science 331, 439 (2011)]. We argue that coupling of this order parameter to magnetic probes can be hidden by crystal-field effects, while still having significant effects on transport, thermodynamics, and magnetic susceptibilities. In a simple tight-binding model for the heavy quasiparticles, we show the connection between the hidden order and antiferromagnetic phases arises since they form different components of this single rank-5 pseudospin vector. Using a phenomenological theory, we show that the experimental pressure-temperature phase diagram can be qualitatively reproduced by tuning terms which break pseudospin rotational symmetry. As a test of our proposal, we predict the presence of small magnetic moments in the basal plane oriented in the [110] direction ordered at the wave vector (0,0,1).}, + number = {24}, + journal = {Physical Review B}, + doi = {10/gf5vbn}, + author = {Rau, Jeffrey G. and Kee, Hae-Young}, + month = jun, + year = {2012}, + pages = {245112}, + file = {/home/pants/.zotero/data/storage/6HP8DPHU/Rau and Kee - 2012 - Hidden and antiferromagnetic order as a rank-5 sup.pdf} +} + +@article{tonegawa_cyclotron_2012, + title = {Cyclotron {{Resonance}} in the {{Hidden}}-{{Order Phase}} of \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, + volume = {109}, + abstract = {We report the first observation of cyclotron resonance in the hidden-order phase of ultraclean URu2Si2 crystals, which allows the full determination of angle-dependent electron-mass structure of the main Fermi-surface sheets. We find an anomalous splitting of the sharpest resonance line under in-plane magnetic-field rotation. This is most naturally explained by the domain formation, which breaks the fourfold rotational symmetry of the underlying tetragonal lattice. The results reveal the emergence of an in-plane mass anisotropy with hot spots along the [110] direction, which can account for the anisotropic in-plane magnetic susceptibility reported recently. This is consistent with the ``nematic'' Fermi liquid state, in which itinerant electrons have unidirectional correlations.}, + number = {3}, + journal = {Physical Review Letters}, + doi = {10/f35jzf}, + author = {Tonegawa, S. and Hashimoto, K. and Ikada, K. and Lin, Y.-H. and Shishido, H. and Haga, Y. and Matsuda, T. D. and Yamamoto, E. and Onuki, Y. and Ikeda, H. and Matsuda, Y. and Shibauchi, T.}, + month = jul, + year = {2012}, + pages = {036401}, + file = {/home/pants/.zotero/data/storage/MHBZ6QTK/Tonegawa et al. - 2012 - Cyclotron Resonance in the Hidden-Order Phase of $.pdf} +} + +@article{thalmeier_signatures_2011, + title = {Signatures of Hidden-Order Symmetry in Torque Oscillations, Elastic Constant Anomalies, and Field-Induced Moments in {{URu}}\$\{\}\_\{2\}\${{Si}}\$\{\}\_\{2\}\$}, + volume = {83}, + abstract = {We discuss the conclusions on the symmetry of hidden order (HO) in URu2Si2 that may be drawn from recent torque experiments in a rotating magnetic field by Okazaki et al. [Science 331, 439 (2011)] (to be published). They are very sensitive to changes in the magnetic susceptibility induced by HO. We show that the observed twofold angular torque oscillations give evidence that HO has degenerate E-type (yz, zx) symmetry where both components are realized. The oscillations have the wrong characteristics or are absent for the one-dimensional (1D) nontrivial representations like quadrupolar B1(x2-y2) and B2(xy) type HO or hexadecapolar A2[xy(x2-y2)] type HO. Therefore, they may be excluded as candidates for HO. We also predict the field-angular variation of possible field-induced Bragg peaks based on the underlying E-type order parameter and discuss the expected elastic constant anomalies.}, + number = {16}, + journal = {Physical Review B}, + doi = {10/bjx43x}, + author = {Thalmeier, Peter and Takimoto, Tetsuya}, + month = apr, + year = {2011}, + pages = {165110}, + file = {/home/pants/.zotero/data/storage/UD5E2IUD/Thalmeier and Takimoto - 2011 - Signatures of hidden-order symmetry in torque osci.pdf} +} + +@article{harima_why_2010, + title = {Why the {{Hidden Order}} in {{URu2Si2 Is Still Hidden}}\textendash{{One Simple Answer}}}, + volume = {79}, + issn = {0031-9015}, + abstract = {For more than two decades, the nonmagnetic anomaly observed around 17.5 K in URu 2 Si 2 , has been investigated intensively. However, any kind of fingerprint for the lattice anomaly has not been observed in the low-temperature ordered phase. Therefore, the order has been called ``the hidden order''. One simple answer to why the hidden order is still hidden is presented from the space group analysis. The second-order phase transition from I 4/ m m m (No.~139) to P 4 2 / m n m (No.~136) does not require any kind of lattice distortion in this system and allows the NQR frequency at a Ru site unchanged. It is compatible with O x y -type antiferro-quadrupole ordering with Q =(0, 0, 1). The characteristics of the hidden order are discussed based on the local 5 f 2 electron picture.}, + number = {3}, + journal = {Journal of the Physical Society of Japan}, + doi = {10/fgmjmf}, + author = {Harima, Hisatomo and Miyake, Kazumasa and Flouquet, Jacques}, + month = mar, + year = {2010}, + pages = {033705}, + file = {/home/pants/.zotero/data/storage/2MY7VK9P/Harima et al. - 2010 - Why the Hidden Order in URu2Si2 Is Still Hidden–On.pdf} +} + +@article{santini_crystal_1994, + title = {Crystal {{Field Model}} of the {{Magnetic Properties}} of {{U}}\$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Ru}}\vphantom\{\}\vphantom\{\}\_\{2\}\$\$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, + volume = {73}, + abstract = {We propose a model based on quadrupolar ordering of localized f electrons to explain the 17.5 K phase transition of URu2Si2. The tiny staggered magnetic moment observed by neutron scattering is interpreted as a weak secondary effect associated to the symmetry-breaking perturbation. The model is able to account for the observed behavior of the linear and nonlinear susceptibilities throughout the transition. A connection with the quadrupolar Kondo theory is proposed.}, + number = {7}, + journal = {Physical Review Letters}, + doi = {10/fn6ntc}, + author = {Santini, P. and Amoretti, G.}, + month = aug, + year = {1994}, + pages = {1027-1030}, + file = {/home/pants/.zotero/data/storage/2ZPUF4NZ/Santini and Amoretti - 1994 - Crystal Field Model of the Magnetic Properties of .pdf} +} + +@article{ohkawa_quadrupole_1999, + title = {Quadrupole and Dipole Orders in {{URu2Si2}}}, + volume = {11}, + issn = {0953-8984}, + abstract = {Exotic magnetism below TN17.5 K is studied within the level scheme where the lowest multiplet is a doublet within the 5f2 configuration. Effective g-factors of pseudo-spins with S = \textonehalf, which describe the degree of freedom of the doublet, are highly anisotropic: gx = gy = 0 for the xy-components and gz0 for the z-component. It is proposed that a recently discovered transition of first order at a critical pressure pc1.5 GPa is that between an ordered state of quadrupoles, with order parameter O(x2)-y2 or Oxy, below pc and an ordered state of dipoles, with order parameter Oz, above pc; pseudo-spins are ordered within the xy-plane below pc, and they are along the z-axis above pc. The proposal of this scenario is followed by many predictions. No static magnetic moments exist below pc. The anisotropy of Van Vleck's susceptibility within the xy-plane is of twofold symmetry corresponding to O(x2)-y2 or Oxy. What one observes by means of neutron diffraction and {$\mathrm{\mu}$}SR (muon spin resonance) below pc are dynamically but slowly fluctuating magnetic moments. The softening of magnons occurs with pressures approaching pc below pc. Although static magnetic moments exist above pc, no magnon excitations can be observed there.}, + language = {en}, + number = {46}, + journal = {Journal of Physics: Condensed Matter}, + doi = {10/bcspzg}, + author = {Ohkawa, Fusayoshi J. and Shimizu, Hirofumi}, + month = nov, + year = {1999}, + pages = {L519--L524}, + file = {/home/pants/.zotero/data/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/data/storage/KAXQ32EJ/Cricchio et al. - 2009 - Itinerant Magnetic Multipole Moments of Rank Five .pdf} +} + +@article{kung_chirality_2015, + title = {Chirality Density Wave of the ``Hidden Order'' Phase in {{URu2Si2}}}, + volume = {347}, + copyright = {Copyright \textcopyright{} 2015, American Association for the Advancement of Science}, + issn = {0036-8075, 1095-9203}, + abstract = {Uncovering the symmetry of a hidden order +Cooling matter generally makes it more ordered and may induce dramatic transitions: Think of water becoming ice. With increased order comes loss of symmetry; water in its liquid form will look the same however you rotate it, whereas ice will not. Kung et al. studied the symmetry properties of a mysteriously ordered phase of the material URu2Si2 that appears at 17.5 K. They shone laser light on the crystal and studied the shifts in the frequency of the light. The electron orbitals of the uranium had a handedness to them that alternated between the atomic layers. +Science, this issue p. 1339 +A second-order phase transition in a physical system is associated with the emergence of an ``order parameter'' and a spontaneous symmetry breaking. The heavy fermion superconductor URu2Si2 has a ``hidden order'' (HO) phase below the temperature of 17.5 kelvin; the symmetry of the associated order parameter has remained ambiguous. Here we use polarization-resolved Raman spectroscopy to specify the symmetry of the low-energy excitations above and below the HO transition. We determine that the HO parameter breaks local vertical and diagonal reflection symmetries at the uranium sites, resulting in crystal field states with distinct chiral properties, which order to a commensurate chirality density wave ground state. +Raman spectroscopy is used to uncover an unusual ordering in the low-temperature phase of a heavy fermion compound. +Raman spectroscopy is used to uncover an unusual ordering in the low-temperature phase of a heavy fermion compound.}, + language = {en}, + number = {6228}, + journal = {Science}, + doi = {10/f6479q}, + author = {Kung, H.-H. and Baumbach, R. E. and Bauer, E. D. and Thorsm{\o}lle, V. K. and Zhang, W.-L. and Haule, K. and Mydosh, J. A. and Blumberg, G.}, + month = mar, + year = {2015}, + pages = {1339-1342}, + file = {/home/pants/.zotero/data/storage/E93SDWTG/Kung et al. - 2015 - Chirality density wave of the “hidden order” phase.pdf}, + pmid = {25678557} +} + +@article{kusunose_hidden_2011, + title = {On the {{Hidden Order}} in {{URu2Si2}} \textendash{} {{Antiferro Hexadecapole Order}} and {{Its Consequences}}}, + volume = {80}, + issn = {0031-9015}, + abstract = {An antiferro ordering of an electric hexadecapole moment is discussed as a promising candidate for the long standing mystery of the hidden order phase in URu 2 Si 2 . Based on localized f -electron picture, we discuss the rationale of the selected multipole and the consequences of the antiferro hexadecapole order of x y ( x 2 - y 2 ) symmetry. The mean-field solutions and the collective excitations from them explain reasonably significant experimental observations: the strong anisotropy in the magnetic susceptibility, characteristic behavior of pressure versus magnetic field or temperature phase diagrams, disappearance of inelastic neutron-scattering intensity out of the hidden order phase, and insensitiveness of the NQR frequency at Ru-sites upon ordering. A consistency with the strong anisotropy in the magnetic responses excludes all the multipoles in two-dimensional representations, such as ( O y z , O z x ). The expected azimuthal angle dependences of the resonant X-ray scattering amplitude are given. The ( x 2 - y 2 )-type antiferro quadrupole should be induced by an in-plane magnetic field along [110], which is reflected in the thermal expansion and the elastic constant of the transverse ( c 11 - c 12 )/2 mode. The ( x 2 - y 2 )-type [( x y )-type] antiferro quadrupole is also induced by applying the uniaxial stress along [110] direction [[100] direction]. A detection of these induced antiferro quadrupoles under the in-plane magnetic field or the uniaxial stress using the resonant X-ray scattering provides a direct redundant test for the proposed order parameter.}, + number = {8}, + journal = {Journal of the Physical Society of Japan}, + doi = {10/csgkg7}, + author = {Kusunose, Hiroaki and Harima, Hisatomo}, + month = jul, + year = {2011}, + pages = {084702}, + file = {/home/pants/.zotero/data/storage/VSG5VAMT/Kusunose and Harima - 2011 - On the Hidden Order in URu2Si2 – Antiferro Hexadec.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$} corresponds to the large-moment antiferromagnetic phase. Together, they provide a unified picture of the two broken-symmetry phases in this material.}, + language = {en}, + number = {11}, + journal = {Nature Physics}, + doi = {10/fw2wcx}, + author = {Haule, Kristjan and Kotliar, Gabriel}, + month = nov, + year = {2009}, + pages = {796-799}, + file = {/home/pants/.zotero/data/storage/L3WFEVLT/Haule and Kotliar - 2009 - Arrested Kondo effect and hidden order in URusub.pdf} +} + +@article{kambe_odd-parity_2018, + title = {Odd-Parity Electronic Multipolar Ordering in \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$: {{Conclusions}} from {{Si}} and {{Ru NMR}} Measurements}, + volume = {97}, + shorttitle = {Odd-Parity Electronic Multipolar Ordering in \$\{\textbackslash{}mathrm\{\vphantom{\}\}}{{URu}}\vphantom\{\}\vphantom\{\}\_\{2\}\{\textbackslash{}mathrm\{\vphantom{\}\}}{{Si}}\vphantom\{\}\vphantom\{\}\_\{2\}\$}, + abstract = {We report 29Si and 101Ru NMR measurements on high-quality, single-crystal URu2Si2 samples with a residual resistivity ratio RRR{$\sim$}70. Our results show that the Si and Ru sites exhibit fourfold electronic symmetry around the c axis in the hidden-order state. A previously observed twofold contribution of Si NMR linewidth is concluded to be due to extrinsic magnetic centers. Since the U and Si sites are aligned along the c axis, we conclude further that the electronic state shows fourfold symmetry around the U site below the hidden-order transition. From this observed local symmetry, possible space groups for the hidden-order state are P4/nnc or I4/m, based on group theoretical considerations. Since the order vector is considered to be Q=(001), the hidden-order state is then found to be P4/nnc with rank 5 odd parity, i.e., electric dotriacontapolar order.}, + number = {23}, + journal = {Physical Review B}, + doi = {10/gf5vbp}, + author = {Kambe, S. and Tokunaga, Y. and Sakai, H. and Hattori, T. and Higa, N. and Matsuda, T. D. and Haga, Y. and Walstedt, R. E. and Harima, H.}, + month = jun, + year = {2018}, + pages = {235142}, + file = {/home/pants/.zotero/data/storage/UQWWD3SU/Kambe et al. - 2018 - Odd-parity electronic multipolar ordering in $ ma.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/dx7d23}, + 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 $mathrm U m.pdf} +} + @@ -1,46 +1,46 @@ -\documentclass[aps,prl,reprint]{revtex4-1} +\documentclass[aps,prl,reprint,longbibliography]{revtex4-1} \usepackage[utf8]{inputenc} \usepackage{amsmath,graphicx,upgreek,amssymb} % Our mysterious boy -\def\urusi{URu$_2$Si$_2\ $} +\def\urusi{URu$_{\text2}$Si$_{\text2}$} -\def\e{{\mathrm e}} % "elastic" -\def\o{{\mathrm o}} % "order parameter" -\def\i{{\mathrm i}} % "interaction" +\def\e{{\text e}} % "elastic" +\def\o{{\text o}} % "order parameter" +\def\i{{\text i}} % "interaction" -\def\Dfh{D$_{4\mathrm h}$} +\def\Dfh{D$_{\text{4h}}$} % Irreducible representations (use in math mode) -\def\Aog{{\mathrm A_{1\mathrm g}}} -\def\Atg{{\mathrm A_{2\mathrm g}}} -\def\Bog{{\mathrm B_{1\mathrm g}}} -\def\Btg{{\mathrm B_{2\mathrm g}}} -\def\Eg {{\mathrm E_{ \mathrm g}}} -\def\Aou{{\mathrm A_{1\mathrm u}}} -\def\Atu{{\mathrm A_{2\mathrm u}}} -\def\Bou{{\mathrm B_{1\mathrm u}}} -\def\Btu{{\mathrm B_{2\mathrm u}}} -\def\Eu {{\mathrm E_{ \mathrm u}}} +\def\Aog{{\text A_{\text{1g}}}} +\def\Atg{{\text A_{\text{2g}}}} +\def\Bog{{\text B_{\text{1g}}}} +\def\Btg{{\text B_{\text{2g}}}} +\def\Eg {{\text E_{\text g}}} +\def\Aou{{\text A_{\text{1u}}}} +\def\Atu{{\text A_{\text{2u}}}} +\def\Bou{{\text B_{\text{1u}}}} +\def\Btu{{\text B_{\text{2u}}}} +\def\Eu {{\text E_{\text u}}} % Variables to represent some representation -\def\X{\mathrm X} -\def\Y{\mathrm Y} +\def\X{\text X} +\def\Y{\text Y} % Units -\def\J{\mathrm J} -\def\m{\mathrm m} -\def\K{\mathrm K} -\def\GPa{\mathrm{GPa}} -\def\A{\mathrm{\c A}} +\def\J{\text J} +\def\m{\text m} +\def\K{\text K} +\def\GPa{\text{GPa}} +\def\A{\text{\c A}} % Other -\def\G{\mathrm G} % Ginzburg +\def\G{\text G} % Ginzburg \begin{document} -\title{Elastic properties of \urusi are reproduced by modulated $\Bog$ order} +\title{Elastic properties of \urusi\ are reproduced by modulated $\Bog$ order} \author{Jaron Kent-Dobias} \author{Michael Matty} \author{Brad Ramshaw} @@ -55,7 +55,7 @@ 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 + 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 @@ -68,7 +68,7 @@ \begin{enumerate} \item Introduction \begin{enumerate} - \item \urusi hidden order intro paragraph, discuss the phase diagram + \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 @@ -86,48 +86,58 @@ \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 theory. However, to construct such a theory, one must know the symmetries -of the order parameter, 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 pd paper], and at sufficiently -large [hydrostatic?] pressures, both give way to local moment antiferromagnetism. -Despite over thirty years of effort, the symmetry of the hidden order state remains unknown, and modern theories -\cite{kambe:pr2018a, haule:np2009a, kusunose:jpsj2011a, kung:s2015a,cricchio:prl2009a,ohkawa:jpcm1999a,santini:prl1994a,kiss:ap2004a,harima:jpsj2010a,thalmeier:pr2011a,tonegawa:prl2012a,rau:pr2012a,riggs:nc2015a,hoshino:jpsj2013a,ikeda:prl1998a,chandra:n2013a,harrison:apa2019a,ikeda:np2012a} -propose a variety of possibilities. -Many [all?] of these theories rely on the formulation of a microscopic model for the -HO state, but without direct experimental observation of the broken symmetry, none -have been confirmed. - -One case that does not rely on a microscopic model is recent work \cite{ghosh:apa2019a} -that studies the HO transition using resonant ultrasound spectroscopy (RUS). -RUS is an experimental technique that measures mechanical resonances of a sample. These -resonances contain information about the full elastic tensor of the material. Moreover, -the frequency locations of the resonances are sensitive to symmetry breaking at an electronic -phase transition due to electron-phonon coupling [cite]. Ref.~\cite{ghosh:apa2019a} uses this information -to place strict thermodynamic bounds on the symmetry of the HO OP, again, independent of -any microscopic model. Motivated by these results, in this paper we consider a mean field theory -of an OP coupled to strain and the effect that the OP symmetry has on the elastic response -in different symmetry channels. Our study finds that a single possible OP symmetry -reproduces the experimental strain susceptibilities, and fits the experimental data well. - -We first present a phenomenological Landau-Ginzburg mean field theory of strain coupled to an -order parameter. We examine the phase diagram predicted by this theory and compare it -to the experimentally obtained phase diagram of \urusi. -Then we compute the elastic response to strain, and examine the response function dependence on -the symmetry of the OP. -We proceed to compare the results from mean field theory with data from RUS experiments. -We further examine the consequences of our theory at non-zero applied pressure in comparison -with recent x-ray scattering experiments [cite]. -Finally, we discuss our conclusions and future experimental and theoretical work that our results motivate. - -The point group of \urusi is \Dfh, and any coarse-grained theory must locally +The study of phase transitions is a central theme of condensed matter physics. +In many cases, a phase transition between different states of matter is marked +by a change in symmetry. In this paradigm, the breaking of symmetry in an +ordered phase corresponds to the condensation of an order parameter (OP) that +breaks the same symmetries. Near a second order phase transition, the physics +of the OP can often be described in the context of Landau-Ginzburg mean field +theory. However, to construct such a theory, one must know the symmetries of +the order parameter, 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 pd paper], and at sufficiently large [hydrostatic?] +pressures, both give way to local moment antiferromagnetism. Despite over +thirty years of effort, the symmetry of the hidden order state remains unknown, +and modern theories \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. + +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]. +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 order parameter. We examine the phase diagram predicted by this +theory and compare it to the experimentally obtained phase diagram of \urusi. +Then we compute the elastic response to strain, and examine the response +function dependence on the symmetry of the OP. We proceed to compare the +results from mean field theory with data from RUS experiments. We further +examine the consequences of our theory at non-zero applied pressure in +comparison with recent x-ray scattering experiments [cite]. Finally, we +discuss our conclusions and future experimental and theoretical work that our +results motivate. + +The point group of \urusi\ is \Dfh, and any coarse-grained theory must locally respect this symmetry. We will introduce a phenomenological free energy density in three parts: that of the strain, the order parameter, and their interaction. The most general quadratic free energy of the strain $\epsilon$ is @@ -230,6 +240,10 @@ The uniform--modulated transition is now continuous. The schematic phase diagrams for this model are shown in Figure \ref{fig:phases}. \begin{figure}[htpb] + \includegraphics[width=\columnwidth]{phase_diagram_experiments} + + \vspace{1em} + \includegraphics[width=0.51\columnwidth]{phases_scalar}\hspace{-1.5em} \includegraphics[width=0.51\columnwidth]{phases_vector} \caption{ @@ -397,14 +411,7 @@ this expression can be brought to the form \mathcal I(\xi_{\perp0} q_*|t|^{-1/4}) \lesssim |t|^{13/4}, \end{equation} -where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I$ -is defined by -\begin{equation} - \mathcal I(x)=\frac1\pi\int_{-\infty}^\infty dy\,\frac{\sin\tfrac y2}y - \bigg(\frac1{1+(y^2-x^2)^2} - -\frac{K_1(\sqrt{1+(y^2-x^2)^2})}{\sqrt{1+(y^2-x^2)^2}}\bigg) -\end{equation} -For large argument, $\mathcal I(x)\sim x^{-4}$, yielding +where $\xi=\xi_0t^{-\nu}$ defines the bare correlation lengths and $\mathcal I(x)\sim x^{-4}$ for large $x$, yielding \begin{equation} t_\G^{9/4}\sim\frac{2k_B}{\pi\Delta c_V\xi_{\parallel0}^2\xi_{\perp0}^5q_*^4} \end{equation} @@ -426,6 +433,6 @@ self-consistent. \end{acknowledgements} \bibliographystyle{apsrev4-1} -\bibliography{hidden_order,library} +\bibliography{hidden_order, library} \end{document} diff --git a/phase_diagram_experiments.pdf b/phase_diagram_experiments.pdf Binary files differnew file mode 100644 index 0000000..46dbf22 --- /dev/null +++ b/phase_diagram_experiments.pdf diff --git a/phases_scalar.pdf b/phases_scalar.pdf Binary files differindex 9203f7f..2f0deff 100644 --- a/phases_scalar.pdf +++ b/phases_scalar.pdf diff --git a/phases_vector.pdf b/phases_vector.pdf Binary files differindex fc4cb37..7d4b466 100644 --- a/phases_vector.pdf +++ b/phases_vector.pdf diff --git a/plots.nb b/plots.nb new file mode 100644 index 0000000..2df6f03 --- /dev/null +++ b/plots.nb @@ -0,0 +1,1609 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* CreatedBy='Mathematica 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