A Crystalline Electric Field

Quantum fluctuations in the effective pseudospin-1/2 model for magnetic pyrochlore oxides

Abstract

The effective quantum pseudospin- model for interacting rare-earth magnetic moments, which are locally described with atomic doublets, is studied theoretically for magnetic pyrochlore oxides. It is derived microscopically for localized Pr moments in PrO (Zr, Sn, Hf, and Ir) by starting from the atomic non-Kramers magnetic doublets and performing the strong-coupling perturbation expansion of the virtual electron transfer between the Pr and O electrons. The most generic form of the nearest-neighbor anisotropic superexchange pseudospin- Hamiltonian is also constructed from the symmetry properties, which is applicable to Kramers ions Nd, Sm, and Yb potentially showing large quantum effects. The effective model is then studied by means of a classical mean-field theory and the exact diagonalization on a single tetrahedron and on a 16-site cluster. These calculations reveal appreciable quantum fluctuations leading to quantum phase transitions to a quadrupolar state as a melting of spin ice for the Pr case. The model also shows a formation of cooperative quadrupole moment and pseudospin chirality on tetrahedrons. A sign of a singlet quantum spin ice is also found in a finite region in the space of coupling constants. The relevance to the experiments is discussed.

pacs:
75.10.Jm, 75.10.Dg, 75.30.Kz, 75.50.Ee

I Introduction

Quantum fluctuations and geometrical frustration are a couple of key ingredients in realizing nontrivial spin-disordered ground states without a magnetic dipole long-range order (LRO) in three spatial dimensions (1); (2); (3); (4). The pyrochlore lattice structure is a typical example where the geometrical frustration plays a crucial role in preventing the LRO (1); (5); (6). Of our particular interest in this paper is a so-called dipolar spin ice (7); (8); (9); (10), such as DyTiO and HoTiO, and related systems. The dipolar spin ice provides a classical magnetic analogue of a cubic water ice (11) and is characterized by the emergent U(1) gauge field mediating the Coulomb interaction between monopole charges (12); (13) as well as the dipolar spin correlation showing a pinch-point singularity (14); (15). Introducing quantum effects to the classical spin ice may produce further nontrivial states of matter. Evidence of quantum effects has recently been observed with inelastic neutron-scattering experiments on the spin-ice related compounds, TbTiO (16); (17); (18), TbSnO (18), and PrSnO (19). Exploiting a weak coupling of the rare-earth magnetic moments to conduction electrons, a chiral spin state (3) has been detected through the anomalous Hall effect (20) at zero magnetic field without magnetic dipole LRO in another related compound PrIrO (21). Vital roles of the planar components have also been experimentally observed in YbTiO and ErTiO (22). Obviously, quantum fluctuations enrich the otherwise classical properties of the spin ice. They may drive it to other states of matter, including quadrupolar states and chiral spin states (23). The aims of this paper are to provide a comprehensive derivation of a realistic effective quantum model for these spin-ice related materials and to understand its basic properties including nontrivial quantum effects.

i.1 Classical dipolar spin ice

Let us briefly review the classical (spin) ice. The low-energy properties of water and spin ices are described by Ising degrees of freedom that represent whether proton displacements (electric dipoles) and magnetic dipoles, respectively, point inwards (“in”) to or outwards (“out”) from the center of the tetrahedron. The interaction among the Ising variables favors nearest-neighbor pairs of “in” and “out” and thus suffers from geometrical frustration. This produces a so-called ice rule (24); (11) stabilizing “2-in, 2-out” configurations on each tetrahedron. Macroscopic degeneracy of this ice-rule manifold produces Pauling’s residual entropy  (11).

In the dipolar spin ice (7); (8); (9), a rare-earth magnetic moment located at a vertex of tetrahedrons plays the role of the Ising variable because of the large crystalline electric field (CEF), which is often approximately modeled by

(1)

with the Landé factor and . Here, is the quantum number for the total angular momentum , and defines a unit vector at a pyrochlore-lattice site that points outwards from the center of the tetrahedron belonging to one fcc sublattice of the diamond lattice and inwards to that belonging to the other sublattice. The amplitude of the rare-earth magnetic moment is so large that the interaction between the magnetic moments is dominated by the magnetic dipolar interaction (25); (26),

(2)

with and the summation over all the pairs of atomic sites. This yields a ferromagnetic coupling  K between the nearest-neighbor magnetic moments for HoTiO and DyTiO with the moment amplitude and the lattice constant  Å (9), providing a main driving force of the ice rule. It prevails over the nearest-neighbor superexchange interaction which is usually assumed to take the isotropic Heisenberg form

(3)

In the limit of , is reduced to an Ising model (26), which can explain many magnetic properties experimentally observed at temperatures well below the crystal-field excitation energy (9); (13); (27); (10).

Because of the ferromagnetic effective nearest-neighbor coupling , creating “3-in, 1-out” and “1-in, 3-out” configurations out of the macroscopically degenerate “2-in, 2-out” spin-ice manifold costs an energy, and can be regarded as defects of magnetic monopoles and anti-monopoles with a unit magnetic charge (6). Then, the spin ice is described as the Coulomb phase of magnetic monopoles where the emergent U(1) gauge fields mediate the Coulomb interaction between monopole charges (12); (13). The density of magnetic monopoles is significantly suppressed to lower the total free energy at a temperature a few Kelvin. Simultaneously, the reduction of the monopole density suppresses spin-flip processes, for instance, due to a quantum tunneling (28), that change the configuration of monopoles. Hence, the relaxation time to reach the thermal equilibrium shows a rapid increase. These phenomena associated with a thermal quench of spin ice have been experimentally observed (29); (30) and successfully mimicked by classical Monte-Carlo simulations on the Coulomb gas model of magnetic monopoles (27); (31). This indicates that the quantum effects are almost negligible in the dipolar spin ice. It has been shown that the emergent gapless U(1) gauge excitations together with a power-law decay of spin correlations can survive against a weak antiferroic exchange interaction that exchanges the nearest-neighbor pseudospin- variables (“in” and “out”) (12). This spin liquid (12) can be viewed as a quantum version of the spin ice, though the macroscopic degeneracy of the ice-rule manifold should eventually be lifted in the ideal case under the equilibrium.

i.2 Quantum effects

At a first glance, one might suspect that quantum fluctuations should be significantly suppressed by a large total angular momentum of the localized rare-earth magnetic moment and its strong single-spin Ising anisotropy , since they favor a large amplitude of the quantum number for , either or . Namely, in the effective Hamiltonian, , a process for successive flips of the total angular momentum from to at one site and from to at the adjacent site is considerably suppressed at a temperature . The coupling constant for this pseudospin-flip interaction is of order of , and becomes negligibly small compared to the Ising coupling .

In reality, however, because of the crystalline electric field (CEF) acting on rare-earth ions [Fig. 1], the conservation of , which is implicitly assumed in the above consideration, no longer holds in the atomic level. Eigenstates of the atomic Hamiltonian including the coupling and the CEF take the form of a superposition of eigenstates of whose eigenvalues are different by integer multiples of three. Obviously, this is advantageous for the quantum spin exchange to efficiently work.

Attempts to include quantum effects have recently been in progress. It has been argued that the presence of a low-energy crystal-field excited doublet above the ground-state doublet in TbTiO (16); (10) enhances quantum fluctuations and possibly drives the classical spin ice into a quantum spin ice composed of a quantum superposition of “2-in, 2-out” configurations (32).

We have proposed theoretically an alternative scenario, namely, a quantum melting of spin ice (23). A quantum entanglement among the degenerate states lifts the macroscopic degeneracy, suppresses the spin-ice freezing, and thus leads to another distinct ground state. Actually, the quantum-mechanical spin-exchange Hamiltonian mixes “2-in, 2-out” configurations with “3-in, 1-out” and “1-in, 3-out”, leading to a failure of the strict ice rule and a finite density of monopoles at the quantum-mechanical ground state. Namely, the quantum-mechanically proliferated monopoles can modify the dipolar spin-ice ground state, while a spatial profile of short-range spin correlations still resembles that of the dipolar spin ice (23). They may appear in bound pairs or in condensates. We have reported that there appears a significantly large anisotropic quantum-mechanical superexchange interaction between Pr magnetic moments in PrO (23) ( = Zr, Sn, Hf, and Ir) (33). This anisotropic superexchange interaction drives quantum phase transitions among the spin ice, quadrupolar states having nontrivial chirality correlations, and the quantum spin ice, as we will see later.

Actually, among the rare-earth ions available for magnetic pyrochlore oxides (33); (10), the Pr ion could optimally exhibit the quantum effects because of the following two facts. (i) A relatively small magnitude of the Pr localized magnetic moment, whose atomic value is given by , suppresses the magnetic dipolar interaction, which is proportional to the square of the moment size. Then, for PrO, one obtains  K, which is an order of magnitude smaller than 2.4 K for HoTiO and DyTiO. Similarly, quantum effects might appear prominently also for Nd, Sm, and Yb ions because of their small moment amplitudes, , , and , respectively, for isolated cases. (ii) With fewer electrons, the -electron wavefunction becomes less localized at atomic sites. This enhances the overlap with the O orbitals at the O1 site [Fig. 1 (a)], and thus the superexchange interaction which is also further increased by a near resonance of Pr and O levels. Moreover, this superexchange interaction appreciably deviates from the isotropic Heisenberg form because of the highly anisotropic orbital shape of the -electron wavefunction and the strong coupling. Since the direct Coulomb exchange interaction is even negligibly small (25), this superexchange interaction due to virtual - electron transfers is expected to be the leading interaction.

Recent experiments on PrSnO (34), PrZrO (35), and PrIrO (36) have shown that the Pr ion provides the Ising moment described by a non-Kramers magnetic doublet. They show similarities to the dipolar spin ice. (i) No magnetic dipole LRO is observed down to a partial spin-freezing temperature -(34); (35); (36); (21); (19); (37). (ii) PrIrO shows a metamagnetic transition at low temperatures only when the magnetic field is applied in the [111] direction (21), indicating the ice-rule formation due to the effective ferromagnetic coupling (21). On the other hand, substantially different experimental observations from the dipolar spin ice have also been made. The Curie-Weiss temperature is antiferromagnetic for the zirconate (35) and iridate (36), unlike the spin ice. The stannate shows a significant level of low-energy short-range spin dynamics in the energy range up to a few Kelvin (19), which is absent in the classical spin ice. Furthermore, the iridate shows the Hall effect at zero magnetic field without magnetic dipole LRO (21), suggesting an onset of a chiral spin-liquid phase (3) at  K.

The discovery of this chiral spin state endowed with a broken time-reversal symmetry on a macroscopic scale in PrIrO without apparent magnetic LRO (21) has increased the variety of spin liquids. One might speculate that this is caused mainly by a Kondo coupling to Ir conduction electrons and thus the RKKY interaction (38). However, the low-temperature thermodynamic properties are common in this series of materials, PrO, except that a small partial reduction () of Pr magnetic moments probably due to conduction electrons affects the resistivity and the magnetic susceptibility in PrIrO (36). Furthermore, the onset temperature  K for the emergent anomalous Hall effect is comparable to the ferromagnetic coupling  K  (21). Therefore, it is natural to expect that a seed of the chiral spin state below exists in the Pr moments interacting through the superexchange interaction and possibly the state is stabilized by the conduction electrons. Another intriguing observation here is that without appreciable quantum effects, the chiral manifold of classical ice-rule spin configurations (21) that has been invented to account for the emergent anomalous Hall effect will result from a magnetic dipole LRO or freezing, which is actually absent down to . This points to a significant level of nontrivial quantum effects.

Figure 1: (Color online) (a) Pr ions form tetrahedrons (dashed lines) centered at O ions (O1), and are surrounded by O ions (O2) in the symmetry as well as by transition-metal ions (). Each Pr magnetic moment (bold arrow) points to either of the two neighboring O1 sites. denotes the local coordinate frame. (b) The local coordinate frame from the top. The upward and the downward triangles of the O ions (O2) are located above and below the hexagon of the ions.

In this paper, we develop a realistic effective theory for frustrated magnets PrO on the pyrochlore lattice and provide generic implications on quantum effects in spin-ice related materials, giving a comprehensive explanation of our recent Letter (23). In Sec. II, the most generic nearest-neighbor pseudospin- Hamiltonian for interacting magnetic moments on the pyrochlore lattice is derived on a basis of atomic magnetic doublets for both non-Kramers and Kramers ions. In particular, it is microscopically derived from strong-coupling perturbation theory in the Pr case. We analyze the model for the non-Kramers case by means of a classical mean-field theory in Sec. III, which reveals spin-ice, antiferroquadrupolar, and noncoplanar ferroquadrupolar phases at low temperatures. Then, we perform exact-diagonalization calculations for the quantum pseudospin- case on a single tetrahedron in Sec. IV and on the 16-site cube in Sec. V. We have found within the 16-site cluster calculations a cooperative ferroquadrupolar phase, which is accompanied by crystal symmetry lowering from cubic to tetragonal and can then be categorized into a magnetic analog of a smectic or crystalline phase (39). This provides a scenario of the quantum melting of spin ice and can explain the experimentally observed magnetic properties, including powder neutron-scattering experiments on PrSnO and the magnetization curve on PrIrO. We also reveal a possible source of the time-reversal symmetry breaking observed in PrIrO. It takes the form of the solid angle subtended by four pseudospins on a tetrahedron, each of which is composed of the Ising dipole magnetic moment and the planar atomic quadrupole moment, and shows a nontrivial correlation because of a geometrical frustration associated with the fcc sublattice structure. A possible sign of a singlet quantum spin-ice state has also been obtained within the 16-site numerical calculations in another finite region of the phase diagram. Sec. VI is devoted to discussions and the summary.

Ii Derivation of the effective model

In this section, we will give a microscopic derivation of the effective pseudospin- Hamiltonian in a comprehensive manner. Though we focus on localized moments of Pr ions, the form of our nearest-neighbor anisotropic pseudospin- Hamiltonian is most generic for atomic non-Kramers magnetic doublets. We will also present the generic form of the nearest-neighbor Hamiltonian for Kramers doublets of Nd, Sm, and Yb.

ii.1 Atomic Hamiltonian for Pr

Coulomb repulsion

The largest energy scale of the problem should be the local Coulomb repulsion among Pr electrons. A photoemission spectroscopy on PrO, which is desirable for its reliable estimate, is not available yet. A typical value obtained from Slater integrals for Pr ions is of order of 3-5 eV (40). A cost of the Coulomb energy becomes , , and for the occupation of one, two, and three electrons, respectively [Fig. 2]. For Pr ions, the O electron level at the O1 site should be higher than the level and lower than the level [Fig. 2]. Then, it would be a reasonably good approximation to start from localized states for Pr configurations and then to treat the other effects as perturbations.

Figure 2: (Color online) Local level scheme for and electrons, and the local quantization axes and .

 coupling for configurations

We introduce operators , , and for the total, orbital, and spin angular momenta of electron states of Pr. Within this manifold, the predominant coupling in gives the ground-state manifold with the quantum numbers , , and for the total, orbital, and spin angular momenta, respectively.

Crystalline electric field

The ninefold degeneracy of the ground-state manifold is partially lifted by the local crystalline electric field (CEF), which has the symmetry about the direction toward the O1 site. We define the local quantization axis as this direction. Then, the Hamiltonian for the CEF,

(4)

contains not only orthogonal components with but also off-diagonal components with and , all of which become real if we take and axes as and depicted in Figs. 1 (a) and (b). Here, and denote the annihilation and creation operators of an electron with the components and of the orbital and spin angular momenta, respectively, in the local coordinate frame at the Pr site . The formal expressions for within the point-charge analysis are given in Appendix A. In the rest of Sec. II.1, we drop the subscript for the site for brevity.

We perform the first-order degenerate perturbation theory, which replaces Eq. (4) with , where is the projection operator onto the manifold. First, let us introduce a notation of for the eigenstate corresponding to the orbital and spin quantum numbers and in the local coordinate frame. It is straightforward to express the eigenstates of in terms of and then in terms of -electron operators,

(5)

as explicitly written in Appendix B.

Finally, we obtain the following representation of Eq. (4) in terms of Eq. (5)

(6)

The CEF favors configurations that are linearly coupled to and because of the CEF. This leads to the atomic non-Kramers magnetic ground-state doublet,

(7)

with small real coefficients and as well as . For PrIrO, the first CEF excited state is a singlet located at 168 K and the second is a doublet at 648 K (41). They are located at 210 K and 430 K for PrSnO (19). These energy scales are two orders of magnitude larger than that of our interest,  K. Hence it is safe to neglect these CEF excitations for our purpose. Then, it is convenient to introduce the Pauli matrix vector for the pseudospin- representing the local doublet at each site , so that Eq. (7) is the eigenstate of with the eigenvalue .

Note that in the case of Tb, the first CEF excited state is a doublet at a rather low energy  K (42), and the effects of the first excited doublet cannot be ignored (32) when a similar analysis is performed. Nevertheless, since this CEF excitation in Tb is an order of magnitude larger than , it could be integrated out (32). Then, the model reduces to a similar form of the effective pseudospin- Hamiltonian which has been derived in Ref. (23) and will also be discussed below, though the explicit form has not been presented as far as we know.

ii.2 Dipole and quadrupole moments

Only the component of the pseudospin contributes to the magnetic dipole moment represented as either “in” or “out”, while the transverse components and correspond to the atomic quadrupole moment, i.e., the orbital. This can be easily shown by directly calculating the Pr magnetic dipole and quadrupole moments in terms of the pseudospin. We first take the projection of the total angular momentum to the subspace of the local non-Kramers magnetic ground-state doublet described by Eq. (7). It yields

(8)
(9)

with . With the Landé factor and the Bohr magneton , the atomic magnetic dipole moment is given by

(10)

Note that cannot linearly couple to neutron spins without resorting to higher CEF levels. On the other hand, the quadrupole moments are given by

(11)

This is a general consequence of the so-called non-Kramers magnetic doublet and not restricted to the Pr ion. Namely, when the atomic ground states of the non-Kramers ions having an even number of electrons and thus an integer total angular momentum are described by a magnetic doublet, only contributes to the magnetic dipole moment, while corresponds to the atomic quadrupole moment. This sharply contrasts to the following two cases: (i) In the case of Kramers doublets, all three components of may contribute to the magnetic dipole moment while their coefficients can be anisotropic. (ii) In the case of non-Kramers non-magnetic doublets, corresponds to a quadrupole moment or even a higher-order multipole moment which is a time-reversal invariant.

ii.3 Superexchange interaction

Now we derive the superexchange Hamiltonian through the fourth-order strong-coupling perturbation theory. Keeping in mind the local level scheme of Pr electrons and O electrons at O1 sites, which has been explained in Sec. II.1, we consider nonlocal effects introduced by the electron transfer between the Pr orbital and the O orbital.

Local coordinate frames

In order to symmetrize the final effective Hamiltonian, it is convenient to choose a set of local coordinate frames so that it is invariant under rotations of the whole system about three axes that include an O1 site and are parallel to the global , , or axes, which belong to the space group of the present pyrochlore system. We can start from the local coordinate frame previously defined in Sec. II.1.3 and in Fig. 1 for a certain site and generate the other three local frames by applying the above three rotations.

For instance, we can adopt

(12a)
for the Pr sites at with ,
(12b)
for the Pr sites at with ,
(12c)
for the Pr sites at with , and
(12d)
for the Pr sites at with ,

where represents a fcc lattice vector spanned by , , and with integers , and is the lattice constant, i.e., the side length of the unit cube. In particular, all the local directions attached to the Pr sites belonging to the tetrahedron centered at the O1 site point inwards, and they satisfy the relation

(13)

Actually, other sets of local coordinate frames which are obtained by threefold and sixfold rotations about yield exactly the same expression for the effective Hamiltonian for Kramers and non-Kramers cases, respectively.

These local coordinate frames are related to the following rotations of the global coordinate frame,

(14a)
(14b)
(14c)
(14d)
(14e)

Note that the coordinate frame for the spins is always attached to that for the orbital space in each case. The rotation of with the orbital and the spin of a single electron takes the form

(15)

- hybridization

The electrons occupying the atomic ground-state doublet, Eq. (7), or holes can hop to the O levels at the neighboring O1 site. Because of the symmetry, the - electron transfer along the local axis is allowed only for the bonding () between and orbitals and the bondings () between and orbitals and between and orbitals in the local coordinate frame defined in Eqs. (12) [Fig. 3 (a)]. Their amplitudes are given by two Slater-Koster parameters (43) and , respectively. Then, the Hamiltonian for the - hybridization reads

(16)

with , , and an index for the two fcc sublattices of the diamond lattice, where represents the annihilation operator of a electron at the O1 site with the orbital and spin quantum numbers and , respectively, in the global coordinate frame. Here, transforms the representation from the global frame for to the local frame for .

Figure 3: (Color online) (a) Two - transfer integrals; between and orbitals, and between / and /. (b) - virtual electron hopping processes. () and in the state represent the number of electrons at the Pr site () and that of electrons at the O1 site.

Strong-coupling perturbation theory

Now we are ready to perform the strong-coupling perturbation expansion in and . Hybridization between these Pr electrons and O electrons at the O1 site, which is located at the center of the tetrahedron, couples states having the local energy with and states having the local energy levels 0 and , respectively [Fig. 2]. Here, the coupling has been ignored in comparison with for simplicity. Creating a virtual hole decreases the total energy by , which is the electron level measured from the level.

First, the second-order perturbation in and produces only local terms. They only modify the CEF from the result of the point-charge analysis with renormalized parameters for the effective ionic charges and radii. Nontrivial effects appear in the fourth order in and . Taking into account the virtual processes shown in Fig. 3 (b), the fourth-order perturbed Hamiltonian in and is obtained as

(17)

ii.4 Effective pseudospin- model

Figure 4: (Color online) The pyrochlore lattice structure. The phase appearing in Eq. (18) takes , , and on the blue, red, green bonds, respectively, in our choice of the local coordinate frames [Eqs. 12].

Next we project the superexchange Hamiltonian Eq. (17) onto the subspace of doublets given by Eq. (7). For this purpose, we have only to calculate for a site the matrix elements of the operators with and , in terms of that is explicitly represented with -electron operators in Appendix B, and then in terms of the atomic doublet , Eq. (7). Then, we finally obtain the effective quantum pseudospin- Hamiltonian;

(18)

with , where represents a vector of the Pauli matrices for the pseudospin at a site . The phase (44) takes , , and for the bonds shown in blue, red, and green colors in Fig. 4, in the local coordinate frames defined in Eq. (12). This phase cannot be fully gauged away, because of the noncollinearity of the magnetic moment directions and the threefold rotational invariance of about the [111] axes. Equation (18) gives the most generic nearest-neighbor pseudospin- Hamiltonian for non-Kramers magnetic doublets of rare-earth ions such as Pr and Tb that is allowed by the symmetry of the pyrochlore system. Note that the bilinear coupling terms of and are prohibited by the non-Kramers nature of the moment; namely changes the sign under the time-reversal operation, while does not.

Figure 5: (Color online) A phase diagram for the sign of the dimensionless Ising coupling , defined through Eq. (19), as functions of and for , , , , and . In each case, is positive in the shaded region, and negative otherwise.

The dependence of the Ising coupling constant on , , and takes the form,

(19)

where contains the dependence on the remaining dimensionless variables . We show the sign of as functions of and for several choices of in Fig. 5. In particular, for which includes a realistic case of , is found to be positive. Since the prefactor in Eq. (19) is positive, the Ising coupling is also positive, namely, antiferroic for pseudospins, in this case. Taking account of the tilting of the two neighboring local axes by , this indicates a ferromagnetic coupling between the physical magnetic moments and provides a source of the ice rule.

The CEF produces two quantum-mechanical interactions in the case of non-Kramers ions; the pseudospin-exchange and pseudospin-nonconserving terms. The ratios and of their coupling constants to the Ising one are insensitive to and but strongly depends on and . Figures 6 (a) and (b) show and , respectively, as functions of that characterizes the CEF for a typical choice of parameters, , , and , in the cases of . Henceforth, we adopt and , following the point-charge analysis of the inelastic powder neutron-scattering data on PrIrO [Ref. (41)]. Actually, these estimates of and lead to the local moment amplitude,

(20)

according to Eq. (10), which reasonably agrees with the experimental observation on PrIrO (36) and PrZrO (35). Then, we obtain and , indicating the appreciable quantum nature. In general, however, the values of and may vary depending on a transition-metal ion and crystal parameters. We note that a finite was not taken into account in the literature seriously.

It is instructive to rewrite the term as

(21)

where we have introduced a two-dimensional vector composed of the planar components of the pseudospin, , and two orthonormal vectors

(22a)
(22b)

Then, it is clear that the sign of can be absorbed by rotating all the pseudospins about the local axes by . Furthermore, in the particular case or , the planar () part, namely, the sum of the and terms, of the Hamiltonian Eq. (18) is reduced to the antiferroic or ferroic pseudospin Hamiltonian (45).

Figure 6: (Color online) The coupling constants (a) and (b) as functions of for several choices of , , , , and . We have adopted , , and .

For Kramers ions such as Nd, Er, and Yb, there appears another coupling constant (46); (47) for an additional interaction term

(23)

whose form has been obtained so that it satisfies the threefold rotational symmetry about axes, i.e., , the mirror symmetry about the planes spanned by and for all the pairs of nearest-neighbor sites and , and the twofold rotational symmetry about