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

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Chapter 3 Octupolar Ordering of Γ 8 Ions 37 Octupolar ordering is spontaneous symmetry breaking, and it is a way to break time reversal invariance without magnetic ordering. This raises a fundamental question. The application of an external magnetic field destroys time reversal invariance. Is spontaneous symmetry breaking by octupolar ordering possible in a finite magnetic field? The answer to the above question is delicate. Whether spontaneous symmetry breaking remains possible depends on the direction of the applied magnetic field. For fields pointing in high-symmetry directions, a second-order octupolar transition remains possible, or it may even split into two consecutive transitions. For non-symmetric field directions, the phase transition is suppressed. We study this problem in the context of the ferro-octupolar model in Sections 3.3 and 3.4. The methods developed here are of importance for the latter Chapters as well. It is of general interest to understand how the magnetic field influences different multipoles. It is possible to consider the problem from two angles. First, the field changes the symmetry of the problem and a new symmetry classification of the order parameters has to be used. Second, one may emphasize that different multipoles get coupled in the presence of an external field. We develop both points of view in considerable detail. Most of the results described in this Chapter were published in [2]. The symmetry argument described in Section 3.4.2 was briefly discussed in [1]. 3.1 Octupolar Moments in the Γ 8 Quartet State The Γ 8 irreducible representation occurs twice in the splitting of the tenfold degenerate J = 9/2 manifold of Np 4+ free ion (Fig. 2.2). In what follows, we construct a lattice model in which each site carries the Γ 1 8 quartet of states. Since the Γ 8 irrep occurs twice, symmetry alone cannot tell us the basis functions. Their detailed form depends on the crystal field potential. However, for many aspects of the problem the specific form of the basis states is not essential, what matters is that they are Γ 8 basis states. For the sake of simplicity, we choose the Γ 8 eigenstates of a purely fourth-order cubic crystal field potential (in standard notations O4 0 + 5O4). 4 The four states represented in terms of the basis |J z 〉 of J = 9/2 are (the numerical coefficients are given in Appendix B): Γ 1 8 = 〉〉〉 7 α ∣ + β ∣ 2 + γ ∣ 2 2 Γ 2 8 = 〉〉〉 9 1 γ ∣ + β ∣ + α ∣ 2 2 2
Chapter 3 Octupolar Ordering of Γ 8 Ions 38 Γ 3 8 = 〉〉 5 δ ∣ + ɛ ∣ 2 2 Γ 4 8 = 〉〉 3 ɛ ∣ + δ ∣ 2 . 2 (3.1) The Γ 8 quartet is composed of two time-reversed pairs, thus it has twofold Kramers, and also twofold non-Kramers degeneracy 1 . We determine the local order parameters supported by the Γ 8 subspace using the method developed in Section 2.3. The decomposition Γ 8 ⊗Γ 8 = Γ 1g ⊕Γ 4u ⊕Γ 3g ⊕Γ 5g ⊕Γ 2u ⊕Γ 4u ⊕Γ 5u (3.2) (where subscripts g and u refer to ”even” and ”odd” parity under time reversal) shows that this subspace supports 15 different kinds of moments: 3 dipoles (Γ 4 ), 5 quadrupoles (Γ 3 and Γ 5 ) and 7 octupoles (Γ 2 , Γ 4 and Γ 5 ). The form of these operators was listed in Table. 2.3. The fourfold local degeneracy will be lifted by the effective fields derived from inter-site interactions. (3.2) shows that there can be 15 different kinds of effective field. The ordering scheme will decide which of these effective fields are non-vanishing. Each of the fields splits the quartet in some way, but not all of them lift the fourfold degeneracy completely. In particular, a quadrupolar effective field cannot resolve the Kramers degeneracy. On the other hand, a dipolar field would not necessarily resolve the non-Kramers degeneracy; this is also true of Γ 2 octupolar field. However, we found that an octupolar field of Γ 5 symmetry can resolve the degeneracy completely, and lead to a unique ground state. We discuss the nature of the effective field arising from the Γ 5u octupoles Tx β , Ty β , and Tz β . It can be rotated in space according to T (ϑ, φ) = sin ϑ(cos φT β x + sin φT β y ) + cos ϑ T β z , (3.3) where ϑ and φ are the standard angles. For a fixed (unit) strength of the octupolar order, the effective field strength varies with orientation as shown in Fig. 3.1 (left). The system chooses an orientation which is energetically preferable. It must also give a unique ground state. As we see, the special orientations Tx β , Ty β and Tz β are in fact, not suitable: the ground state of the corresponding on-site mean-field Hamiltonians has twofold degeneracy. Fig. 3.1 shows that the optimal orientation can be sought in the φ = π/4 plane which contains the (001) and (111) directions. As shown in Fig. 3.1 (right), the ground state is always a singlet except for the special points 2 1 The fact that Kramers degeneracy is present in the system follows from that we have odd number of electrons. See Appendix A for the details. 2 These points correspond to the pure Tz β case discussed above.
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Appendix A Appendix Stevens Operato
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Appendix B Appendix Numerical Param
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Appendix C Appendix Here I list som
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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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