Theme of the Workshop on Itinerant-Electron Magnetism, and Spin Fluctuations

The international workshop on itinerant-electron magnetism was held during September 25-27, 2015 in the seminar house of Graduate School of Science, Kyoto University, Kyoto, Japan. Here, I explain the theme of this workshop, and stress the development of itinerant-electron magnetism in several decades. The workshop was also organized in commemoration of Professor Yoshinori Takahashi’s retirement from University of Hyogo, Japan. Here, I also explain some of his works contributing to the development of itinerant magnetism.


Itinerant-Electron Magnetism and Spin Fluctuations
In these several decades, a lot of important theoretical and experimental approaches have been conducted for understanding the itinerant-electron magnetism [1-20, and references in them]. Among them epoch-making was the great success of the spin-fluctuation theory for weak itinerant ferromagnets and antiferromagnets by Toru Moriya and his coworkers based on the self-consistent renormalization (SCR) of spin fluctuations to magnetic free energy since 1973 (the SCR theory) [6,[8][9][10][11], exceeding the Stoner mean field theory [1] and the dynamical mean field theory, called the random phase approximation (RPA) theory with no mode-mode coupling of spin fluctuations [5]. Afterwards, the spin fluctuation theory has been developed toward the unified theory between the weakly itinerant regime [6] and the localized moment regime [4] in metallic magnets by a phenomenological method by T. Moriya and Yoshinori Takahashi (1978) [8]. Furthermore, the SCR theory was successfully extended to explain the characteristic magnetic behaviors of heavy-fermion systems [17]. Then, the SCR theory has been developed and rearranged in a quantitative way by  [12], by which we can compare the experiments and the SCR theory quantitatively by means of a set of (several numbers of) spin-fluctuation parameters [12,15,16,[18][19][20].

SCR Theory of Spin Fluctuations
In the SCR theory of spin fluctuations for itinerant-electron magnetic systems [6, 8-14, 19, 20], the definitions of spin fluctuations, S 2 = S 2 , which are magnetic excitations and band fluctuations in itinerant-electron magnetic systems, are where S 2 T is the thermal spin fluctuations, S 2 Z.P.
is the zero-point spin fluctuations, q is the wave vector of the spin fluctuation, ω is the frequency of the spin fluctuation, N 0 is the number of magnetic atoms in the system, and Imχ(q, ω) is the imaginary part of the dynamical magnetic susceptibility as a function of q and ω. Furthermore, n(ω) is the bosonic factor. The difference of is only the presence of n(ω) [13,19,20]. The SCR theory of spin fluctuations, which is the mode-mode coupling theory between different wave q-vectors of spin fluctuations, has explained many experimental magnetic properties of the itinerant ferromagnets and antiferromagnets, for example, the low ferromagnetic transition temperature, called the Curie temperature T C , and the low antiferromagnetic transition temperature, called Néel temperature T N , in itinerant-electron systems and the dynamical measurements of spin dynamics in itinerant magnets [6,[8][9][10][11][12][13][14][15][16][17][18][19][20]. The Curie-Weiss law in χ above T C was successfully explained by the T-linear increase of S 2 T in the SCR theory [6,[9][10][11][12]14].
In the SCR theory of spin fluctuations, Imχ(q, ω) is usually written by the following double Lorentzian formula as [9-14, 19, 20] Im χ q,ω where κ and Γ q represent the q (momentum)-and ω (energy)-widths of the spin fluctuation spectrum, and κ 2 corresponds to the inverse susceptibility, 1/χ. In the SCR theory, κ and Γ q are important spin-fluctuation quantities for expressing magnetic quantities. Furthermore, Γ 0 and A are the parameters representing ω -width and q-damping width of the double Lorentzian form. Here, by utilizing a set of spin fluctuation parameters, p s , F 1 , T 0 and T A , magnetic quantities, such as T C , and the temperature dependence of the magnetic susceptibility above T C , can be expressed quantitatively by the SCR theory [9][10][11][12][13][14][15][16][18][19][20], where p s is the spontaneous Bohr magneton number, F 1 , which can be obtained experimentally from the Allott-plot (M 2 vs H/M plot (M being magnetization and H being magnetic field)), is the coefficient of the M 4 -term in Landau expansion of magnetic free energy, and T 0 and T A are the characteristic temperatures corresponding to the q-and ω -widths of the spin fluctuation spectrum deduced from Γ 0 and A as where q B is the wave vector q at the Brillouin zone boundary. For example, the SCR theory gives T C through the following SCR relation among spin-fluctuation parameters as where c is the constant being 0.3353... [12].

Takahashi's Theory of Spin Fluctuations
After providing quantitative evidence for the SCR theory in 1985 [12], Takahashi has developed the spin-fluctuation theory [13] in different approaches with some assumptions: total spin-fluctuation amplitude conservation (TAC) and global consistency (GC) for a few decades (1986 ~ present) [13,19,20], which lead us to the new unified picture of metallic magnetism with a wide variety of itinerant-electron magnets based upon the spin-fluctuation approaches [19,20].
First, he assumed that the following TAC equation (5) is valid even in explaining magnetic properties at finite temperatures [13]. Therefore, total spin fluctuations, namely, the total square amplitude of the local spin fluctuation, are constant and conserved even in the itinerant system as [13,19,20]  = const. (5) Equation (5) is naturally satisfied in the localized moment system. Takahashi assumed the eq. (5) is valid even in an itinerant system, although that is not intuitive in the itinerant system. By using the relation deduced from the eq. (5), the magnetic properties at finite temperatures can be reproduced and explained [13,15,16,[18][19][20][21][22][23]. This allows the new unification of the itinerant-electron magnetism [13,19,20] from the Pauli paramagnetic weak limit [6] to the localized-moment limit, even in the metallic state [4], leading the unified relation between p s /p eff (paramagnetic Bohr magneton number obtained from the Curie constant above T C ) and T C /T 0 independent of magnetic materials as which gives the new universal picture (the universal p eff /p s vs T C /T 0 plot) among various ferromagnetic materials [13,19,20,22] instead of phenomenological Rhodes-Wohlfarth plot (p eff /p s vs T C plot), which was discussed based on the mean field theory [7]. Next, Takahashi has assumed the global consistency (GC) at T C [13]. In the Stoner and 3 SCR theories assumed the Landau expansion up to the M 4 -term is important in magnetic free energy, there occurred the discontinuity at T C in these theories. He has assumed that the M 6 -term is important at T C , resulting in the continuity in M at T C (GC) [13,19,20]. By this assumption he obtained the following relation at T C as

Exotic Superconductivity and Spin Fluctuations
Since the novel superconductors have been discovered in the strongly correlated electron systems, such as heavy-fermion compounds and intermetallics [24], the organic systems [25], the high-Tc cuprates [26,27], and Fe pnictides [28,29], the correlations and interplays between the itinerant magnetism and the novel superconductivity, called exotic superconductivity, have been important, and the itinerant-electron characteristics have recently become one of the most difficult and important problems in the solid state sciences [30][31][32][33]. The formalism of the BCS mean field theory [34][35][36] should be valid even in high-T c cuprate and iron pnictides superconductors, as well as other strongly correlated electron superconductors, such as heavy-fermion superconductors and organic superconductors, although the mediation mechanism of Cooper pairs may be different from that of BCS theory [30][31][32][33]. In high-T c cuprates, microscopic experiments have shown that the magnetic excitations were crucial [37], leading to the possible mechanism involving magnetic interaction-mediated Cooper pairs [30][31][32][33]. In the spin-fluctuation theory, the superconducting transition temperatures T c are found to be universally scaled by the characteristic temperature corresponding to the energy width of spin fluctuations, T 0 , in exotic superconductors [32].

International Workshop on Itinerant-Electron Magnetism (IWIEM)
In this workshop on itinerant-electron magnetism, we had a plan to bring together an international group of leading theoreticians and experimental scientists on magnetism to discuss advanced topics in condensed matter physics, especially related to spin fluctuations in itinerant-electron magnetism including exotic superconductivity with magnetic origins, and to shape the future development of this field. We also planned to invite young scientists as well as graduate students. We hoped that such young scientists had chances to talk with invited speakers and organizers on their own interests.
Finally, this workshop was organized for a celebration of and in commemoration of Professor Yoshinori Takahashi's retirement from University of Hyogo, Japan.

Invited Presentations of IWIEM
The invited talks (speakers and titles) in this workshop were as follows: