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We develop a Clifford algebra approach for 3D Ising model. By utilizing some mathematical facts of the direct product of matrices and their trace, we expand the dimension of the transfer matrices V of the 3D Ising system by adding unit matrices I (with compensation of a factor) and adjusting their sequence, which do not change the trace of the transfer matrices V (Theorem I: Trace Invariance Theorem). It allows us to perform a linearization process on sub-transfer-matrices (Theorem II: Linearization Theorem). It is found that locally for each site j, the internal factor Wj in the transfer matrices can be treated as a boundary factor, which can be dealt with by a procedure similar to the Onsager-Kaufman approach for the boundary factor U in the 2D Ising model. This linearization process splits each sub-transfer matrix into 2n sub-spaces (and the whole system into 2nl sub-spaces). Furthermore, a local transformation is employed on each of the sub-transfer matrices (Theorem III: Local Transformation Theorem). The local transformation trivializes the non-trivial topological structure, while it generalizes the topological phases on the eigenvectors. This is induced by a gauge transformation in the Ising gauge lattice that is dual to the original 3D Ising model. The non-commutation of operators during the processes of linearization and local transformation can be dealt with to be commutative in the framework of the Jordan-von Neumann-Wigner procedure, in which the multiplication in Jordan algebras is applied instead of the usual matrix multiplication AB (Theorem IV: Commutation Theorem)., Comment: 48 pages, 0 figure

We have investigated the elastic properties of the cubic dense Kondo compound Ce0.75La0.25B6 by means of ultrasonic measurements. We have obtained magnetic fields vs temperatures (H-T) phase diagrams under magnetic fields along the crystallographic [001], [110] and [111] axes. An ordered phase IV showing the elastic softening of c44 locates in low temperature region between 1.6 and 1.1 K below 0.7 T in all field directions. The phase IV shows an isotropic nature with regard to the field directions, while the antiferro-magnetic phase III shows an anisotropic character. A remarkable softening of c44 and a spontaneous trigonal distortion εyz+εzx+εxy recently reported by Akatsu et al. [J. Phys. Soc. Jpn. 72 (2003) 205] in the phase IV favor a ferro-quadrupole (FQ) moment of Oyz+Ozx+Oxy induced by an octupole ordering., Comment: 9 figures, Strongly Correlated Electrons

We performed ultrasonic measurement on the rare-earth intermetallic compound PrInAg_2 to examine the quadrupolar Kondo effect associated with the non-Kramers Gamma_3 doublet ground state. The characteristic softening of the elastic constant (c_{11}-c_{12})/2 below 10 K in PrInAg_2 is attributed to a Curie term in quadrupolar susceptibility for the quadrupole O_2^2=J_x^2-J_y^2 of the stable Gamma_3 ground state. (c_{11}-c_{12})/2 turns to a slight increase with the -lnT dependence below 0.1 K, which suggests the quenching of the quadrupolar moment in the quadrupolar Kondo state. Under applied magnetic fields of 10 T and 15 T above 8.7 T corresponding to the Kondo temperature T_K of ~ 0.86 K, the behavior of (c_{11}-c_{12})/2 is described in terms of quadrupolar susceptibility for the stable 4f^2 state., Comment: PDF, 10pages + 5figures, Strongly Correlated Electrons

A 1/2 magnetization plateau in magnetic fields above 23 T and antiferromagnetic (AF) long-range order (AFLRO) in low fields were found in Cu$_2$CdB$_2$O$_6$. Experimental results agree with quantum Monte Carlo results for an expected spin system. There are two kinds of Cu sites [Cu(1) and Cu(2)], which are located adjacent to each other. Unexpectedly, spins on the Cu(1) and Cu(2) sites are in a nearly spin-singlet state and form AFLRO, respectively, although interactions between the Cu(1) and Cu(2) spins cannot be ignored. Cu$_2$CdB$_2$O$_6$ is the first material which shows such coexistence in an atomic scale., Comment: 5 pages, 4 figures

The magnetic phase diagram for antiferro-quadrupole (AFQ) ordering in tetragonal HoB$_2$C$_2$ has been investigated by measurements of elastic constants $C_{11}$, $C_{44}$ and $C_{66}$ in fields along the basal $x$-$y$ plane as well as the principal [001]-axis. The hybrid magnet (GAMA) in Tsukuba Magnetic Laboratory was employed for high field measurements up to 30 T. The AFQ phase is no longer observed above 26.3 T along the principal [001] axis in contrast to the relatively small critical field of 3.9 T in fields applied along the basal [110] axis. The quadrupolar intersite interaction of $O_{xy}$ and/or $O_2^2$ is consistent with the anisotropy in the magnetic phase diagram of the AFQ phase in HoB$_2$C$_2$., Comment: Phys. Rev. B. (2005) in press. approx 8 pages, 10 figures

We present tunneling and rattling motions of an off-center guest atom in cage referring to a clathrate crystal La_3Pd_{20}Ge_6. The elastic constant C_{44} of La_3Pd_{20}Ge_6 shows a Debye-type dispersion around 20 K obeying a relaxation time tau = tau_0exp(E/k_{B}T) with an attempt time tau_0 = 2.0*10^{-12} sec and an activation energy E = 197 K. At low temperatures below 3 K down to 20 mK, the C_{44} shows a softening of C_{44} = C_{44}^0(T-T_C^0)/(T-Theta) with T_C^0 = -337.970 mK and Theta = -338.044 mK. These facts are attributed to two different types of the off-center motions with Gamma_5 symmetry in 4a-site cage of La_3Pd_{20}Ge_6, a thermally activated rattling motion over the potential hill and a tunneling motion through the potential hill at low temperatures., Comment: 5 pages, 4 figures, to be published Phys. Rev. B

We have investigated magnetic properties of Rb$_2$Cu$_2$Mo$_3$O$_{12}$ powder. Temperature dependence of magnetic susceptibility and magnetic-field dependence of magnetization have shown that this cuprate is a model compound of a one-dimensional spin-1/2 Heisenberg system with ferromagnetic first-nearest-neighbor (1NN) and antiferromagnetic second-nearest-neighbor (2NN) competing interactions (competing system). Values of the 1NN and 2NN interactions are estimated as $J_1 = -138$ K and $J_2 = 51$ K ($\alpha \equiv J_2 / J_1 = -0.37$). This value of $\alpha$ suggests that the ground state is a spin-singlet incommensurate state. In spite of relatively large $J_1$ and $J_2$, no magnetic phase transition appears down to 2 K, while an antiferromagnetic transition occurs in other model compounds of the competing system with ferromagnetic 1NN interaction. For that reason, Rb$_2$Cu$_2$Mo$_3$O$_{12}$ is an ideal model compound to study properties of the incommensurate ground state that are unconfirmed experimentally., Comment: 6 pages, 4 figures

Lattice effects in a cerium based clathrate compound Ce_3Pd_20Ge_6 with a cubic Cr_23C_6-type structure have been investigated by ultrasonic and thermal expansion measurements. Elastic softenings of (C_11-C_12)/2 and C_44 proportional to the reciprocal temperature 1/T above T_Q1 = 1.25 K are well described in terms of the quadrupole susceptibility for the ground state Gamma_8 quartet. A huge softening of 50 % in (C_11-C_12)/2 and a spontaneous expansion DL/L = 1.9x10^-4 along the [001] direction in particular indicate the ferroquadrupole ordering of O_2^0 below T_Q1. The elastic anomalies associated with the antiferromagnetic ordering at T_N2 = 0.75 K and the incommensurate antiferromagnetic ordering are also found. Notable frequency dependence of C_44 around 10 K is accounted for by the Debye-type dispersion indicating a Gamma_5 rattling motion of an off-center Ce ion along the [111] direction with eight fractionally occupied positions around the 4a site in a cage. The thermally activated Gamma_5 rattling motion obeying a relaxation time t = t_0exp(E/k_BT) with an attempt time t_0 = 3.1x10^-11 sec and an activation energy E = 70 K dies out with decreasing temperature, and then the off-center tunneling state of Ce ion in the 4a-site cage will appear at low temperatures., Comment: 11 pages, 15 figures, to be published on Phys. Rev. B