Ph.D. THESIS Multipolar ordering in f-electron systems

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Chapter 2 Overview of the Theoretical Background 33 tibility χ 3 . In fact, URu 2 Si 2 was found to develop a weak antiferromagnetic moment ≤ 0.03µ B at, or at least near T N . However, this weak magnetism certainly cannot be the order parameter of the transition: it could not account for the scale of the phenomenon (the ∼ O(ln2) entropy change). Figure 2.7: Left: The linear and the non-linear susceptibility of URu 2 Si 2 as a function of the temperature [56], [65] Right: The specific heat as a function of the temperature. Large anomaly at T = 17.5K indicates the phase transition [65]. The search for the non-magnetic primary order parameter began decades ago. Though (as we have discussed) there are quite a few f-electron systems with difficult-to-identify order parameters, the case of URu 2 Si 2 became known as perhaps the most famous case of ”hidden” order. Though there were many attempts to identify it, neither of the proposed solutions seemed to satisfy all the criteria. In Chapter 5 of the present thesis, I describe the latest attempt to explain the nature of URu 2 Si 2 . As to the weak antiferromagnetism, earlier observations of its samplequality-dependence gave rise to the doubt that it may be extrinsic. Much of recent experimental evidence points to the fact that it is only a minority phase of the samples which is antiferromagnetic. We take this point of view. It follows that the hidden primary order is completely non-magnetic, and at zero field, it does not induce any kind of magnetism. U 4+ : 5f 2 shells are isoelectronic with 4f 2 . In this sense, the problem of U systems has a lot of parallels to that of Pr systems. However, the total number of different ordering schemes supported by J = 4 is huge. URu 2 Si 2 is completely different from the Pr systems mentioned in the previous subsection. The crystal field acting on the U 4+ ion is tetragonal, so the ninefold degeneracy is split ion contributions to χ.
Chapter 2 Overview of the Theoretical Background 34 according to Γ J=4 = 2A 1 ⊕ A 2 ⊕ B 1 ⊕ B 2 ⊕ 2E (2.34) into 7 levels (5 singlets and 2 doublets). Since crystal field theory does not say anything about the order of the levels, and their energy separations, there are many different possibilities. Available experimental evidence was extensively discussed to narrow down the choices. There is general agreement that the crystal field ground state is a singlet. This in itself says that the order (whatever it is) must be of the induced-moment kind: inter-site interaction couple several low-lying levels, and mix out the order parameter. Thus the local Hilbert space is spanned by several crystal field states: the question is, which. Experiments say that at least another two singlets are low-lying, and that any successful fit to the susceptibility up to T ∼ 300K must use at least five crystal field states. It may be the five singlets, or three singlets and a doublet. Earlier schemes tended to ascribe a major role to the quadrupolar degrees of freedom which arise from 3 singlets. A typical level scheme can always be tuned to give a number of reasonable fits, first of all the susceptibility (Fig. 2.7). The reason for not accepting it is that the phase transition of URu 2 Si 2 is definitely not quadrupolar. This was shown by a crucial recent experiment [63]: uniaxial strain in [100] and [110] directions induces largeamplitude antiferromagnetism, while strain in [001] direction does not. Strain is time reversal invariant, so it cannot create time-reversal-invariance breaking order (magnetism). It can only make a preexisting time-reversalinvariance breaking order visible by a mode-coupling effect. Consequently, the hidden order of URu 2 Si 2 must be time-reversal-invariance breaking. Since it is known that it cannot be ordinary magnetism, it must be 22 either octupolar, or an odd-order multipole of still higher order. In Chapter 5, I describe a new model of URu 2 Si 2 , which postulates that the hidden order is octupolar. To support the Tz β order parameter, a new crystal field scheme had to be devised. We show that this model is at least as compatible with standard experiments (Fig. 2.7) as previous ones. It is more important that we can explain strain-induced antiferromagnetism. We mention that recent high-field experiments fully mapped the boundary of the hidden-order phase, and established the existence of a separate highfield phase (Fig. 2.8). Our model can reproduce the essential features of the 22 We think in terms of local on-site order parameters. There are inter-site order parameters with time-reversal-invariance breaking character which have been considered for URu 2 Si 2 : inter-site triple-spin correlators [71], and non-conventional density waves carrying plaquette current [74]. Our model is different from either of these.
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Page 3 and 4: CONTENTS 2 4 PrFe 4 P 12 Skutterudi
Page 5 and 6: Chapter 1 Introduction 4 the orderi
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Page 37 and 38: Chapter 3 Octupolar Ordering of Γ
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Chapter 5 URu 2 Si 2 System We desc
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Chapter 5 URu 2 Si 2 System 86 with
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Chapter 5 URu 2 Si 2 System 88 ∆
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Chapter 5 URu 2 Si 2 System 90 fiel
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Chapter 5 URu 2 Si 2 System 92 Ther
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Chapter 5 URu 2 Si 2 System 94 As w
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Chapter 5 URu 2 Si 2 System 96 8 T=
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Chapter 6 Conclusion One of the aim
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Chapter 6 Conclusion 100 ordering.
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Appendix A Appendix Stevens Operato
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APPENDIX A. APPENDIX 104 becomes as
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Appendix B Appendix Numerical Param
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Appendix C Appendix Here I list som
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Acknowledgement I would like to exp
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BIBLIOGRAPHY 112 [15] P. Santini an
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BIBLIOGRAPHY 114 [51] F. Bourdarot,
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Ph.D. THESIS Multipolar ordering in f-electron systems

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