坂井 徹

The field-induced long-range antiferromagnetic order in the quasi-ID bond-alternating spin system is investigated with the mean field approximation for interchain interaction. The obtained phase diagram in the H-T plane indicated no enhancement of the Neel temperature around the critical fields, although the critical exponent eta characterizing the Tomonaga-Luttinger liquid behavior indicates the antiferromagnetic spin correlation is stronger than at intermediate fields. The experimental phase diagram of F5PNN is well reproduced. (C) 2004 Elsevier B.V. All rights reserved.

Recently, the organic compound biphenyl (BIP-TENO) was synthesized and found to be the first S = 1 spin ladder. The present high-field measurement up to 70 T indicates that a plateau appears in the magnetization curve at M = M(s)/4, where Ms is the saturation magnetization. We propose possible mechanisms of the M = M(s)/4 plateau in BIP-TENO. The frustration in the first, second and third neighbor antiferromagnetic exchange interactions in the S = 1 ladder yields the M = M(s)/4 plateau, with a spontaneous breaking of the translational symmetry. This can be known by use of the degenerate perturbation theory from the strong rung coupling limit. By use of the numerical diagonalization method, we revealed that the most realistic mechanism of the M = M(s)/4 plateau in BIP-TENO is due to the third neighbor interaction. Further theoretical analyses suggest the present mechanism should also lead to a cusp in the magnetization curve. The cusp-like anomaly in the experimental magnetization curve is a considerable evidence for our mechanism. We also discuss the M(s)/2 plateau problem. We show that a small rung coupling cannot yield the M(s)/2 plateau, which is quite different from the M = 0 plateau problem of the S = 1/2 ladder. 2 (C) 2004 Elsevier B.V. All rights reserved.

The field-induced long-range antiferromagnetic order in the quasi-one dimensional S = 1 antiferromagnet is investigated with the exact diagonalization of finite chains and the mean field approximation for the interchain interaction. With increasing external field perpendicular to the principal axis of the single-ion anisotropy, the Haldane gap vanishes at the critical field H-c and the quantum phase transition belongs to the Ising universality class. It is revealed that the gap re-opens for H > H-c, where the long-range antiferromagnetic order appears in the presence of the interchain interaction. The behavior of the gap is consistent with the recent inelastic neutron scattering experiment on NDMAP. (C) 2003 Elsevier B.V. All rights reserved.

We study, at T = 0, the anomalies in the magnetization curve of the S = 1 two-leg ladder with frustrated interactions. We focus mainly on the existence of the M = M-s/2 plateau, where M-s is the saturation magnetization. We report the results using the degenerate perturbation theory and the density matrix renormalization group, which lead to consistent conclusion with each other. We also touch on the M = M-s/4 and (3/4)M-s plateaux and cusps.

We consider a possible mechanism of the charge stripes in the high-T-c cuprates, based on some additional interactions in the two-dimensional t-J model; the next-nearest-neighbour and/or four-spin (ring) exchange interactions. The many-hole correlation functions obtained by numerical exact diagonalization of a finite-cluster t-J model including the correction terms indicate the realization of the mechanism and give some preliminary phase diagrams. A realistic combination of these additional terms is also studied.

The direct electron spin resonance absorption between a singlet ground state and the triplet excited states has been observed in some recent experiments on gapped spin systems, although the transition is forbidden because of the spin conservation law. As possible mechanisms to bring about such a transition, we investigate the effective staggered field due to the alternating g-tensor and the Dzyaloshinskii-Moriya interaction, which have some off-diagonal matrix elements between the singlet and triplet states. The selection rules of the direct transition, which depend on the relative angles among the external field, the polarization of the insident submillimeter wave and the crystal axies, were presented for the two mechanisms in our previous works. We review the selection rules and apply them for the recent experiments on the Haldane chain Ni(C2H8N2)(2)NO2(ClO4), abreviated NENP, the spin Peierls compound CuGeO3 and the Shastry-Sutherland system SrCu2(BO3)(2), to specify the origin of the direct transition.

We propose the spin-Peierls mechanism for the non-trivial magnetization plateaux at M-s/4 and (3/4)M-s (M-s is the saturation magnetization) of the two-leg S = 1 spin ladder, bearing in mind the recent experiment on the BIP-TENO. (C) 2003 Elsevier Science B.V. All rights reserved.

We show a clear experimental evidence of the 1/4 plateau in the high-field magnetization measurement on the first-synthesized S = 1 organic spin ladder compound BIP-TENO. The theoretical mechanism of the plateau formation is also proposed, based on the frustrated interactions. (C) 2003 Elsevier Science B.V. All rights reserved.

The experimental magnetization curve of the first synthesized S = 1 organic spin ladder compound BIP-TENO is presented. It reveals a clear plateau at 1/4 of the saturation magnetization. We propose a mechanism of the 1/4 plateau formation based on the frustration due to third-neighbor exchange interactions. We also compare the numerically calculated magnetization curve and susceptibility with the experimental results to estimate the exchange coupling constants.

Motivated by recent chemical explorations into organic-radical-based higher-spin ladder systems, we study the ground-state properties of a wide class of antiferromagnetic spin-1 ladders. Numerical analysis featuring the level-spectroscopy technique reveals the rich phase diagram, correcting a preceding nonlinear-sigma-model prediction, A variational analysis well interprets the phase competition with particular emphasis on the re-entrant phase boundary on the way from single to coupled chains.

The magnetization curve of the (S,s)=(1,1/2) ferrimagnetic alternating spin chain with single-ion anisotropy D is investigated with the numerical exact diagonalization of finite clusters and size-scaling analyses. The system has a plateau at (1/3) of the saturation moment, which corresponds to the spontaneous magnetization for D=0. Varying D in the 1/3-magnetized ground state under the external field along the axis of D, a quantum phase transition is revealed to occur at the critical value D/J=1.114±0.001 where the plateau vanishes. Except for the critical point, the plateau is always opening, but the mechanism is different between D<Dc and D>Dc. The change of mechanisms is an evidence to clarify that the plateau originates from the quantization of magnetization.

The magnetization curve of the (formula presented) ferrimagnetic alternating spin chain with single-ion anisotropy D is investigated with the numerical exact diagonalization of finite clusters and size-scaling analyses. The system has a plateau at (1/3) of the saturation moment, which corresponds to the spontaneous magnetization for (formula presented) Varying D in the 1/3-magnetized ground state under the external field along the axis of D, a quantum phase transition is revealed to occur at the critical value (formula presented) where the plateau vanishes. Except for the critical point, the plateau is always opening, but the mechanism is different between (formula presented) and (formula presented) The change of mechanisms is an evidence to clarify that the plateau originates from the quantization of magnetization. © 2002 The American Physical Society.

In response to recent attempts to construct higher-spin ladder materials from organic polyradicals, we study the ground-state properties of a wide class of antiferromagnetic spin-1 ladders. Employing various numerical tools, we reveal the rich phase diagram and correct a previous nonlinear-sigma-model prediction. A variational analysis interprets the phase competition well with particular emphasis on the re-entrant phase boundary as a function of the rung interaction.

The electron spin resonance corresponding to the spin gap has been observed in some recent experiments on gapped spin systems, also the direct transition between the singlet ground state and the triplet excited state. One of important origins of the transition is supposed to be the effective staggered field due to some alternations of the g-tensor. We investigate some models of gapped spin systems and obtain a theoretical selection rule for the direct transition in presence of the staggered field. The main results are as follows: (i) When the staggered field is parallel to the external field H, the transition occurs for the magnetic field of the rf-field h perpendicular to H. (ii) When the staggered field is perpendicular to H, it occurs for h parallel to H. In terms of the selection rule, we discuss recent experimental results for the spin Peierls; system CuGeO3.

Very recently, a non-trivial magnetization plateau at 1/4 of the saturation magnetization was observed in the S = 1 spin ladder 3,3′,5,5′-tetrakis(N-tert-butylaminoxyl)biphenyl. In our previous work, we proposed a possible mechanism of the plateau based on the second- and third-neighbor exchange couplings, which lead to frustration. In order to confirm the realization of the mechanism, we compare the temperature dependence of the magnetic susceptibility and some critical magnetic fields obtained by the numerical calculation for the proposed model with the experimental results. © 2002 Elsevier Science Ltd. All rights reserved.

The pseudogap phenomena observed on cuprate high-temperature superconductors are investigated based on the finite-temperature Lanczos method on the 16 sites 4 × 4 cluster t-J model, which includes two holes. The results clearly show the presence of the gap-like behavior in the temperature dependence of various magnetic properties the NMR relaxation rate, the neutron scattering intensity and the static susceptibility. The effect of the external magnetic field on the pseudogap is also studied on the same model. These results are in agreement with our previous exact diagonalization studies on the one-hole system, and confirm our proposal of the predominant role of the antiferromagnetic spin correlation on the observed pseudogap behaviors. © 2002 Elsevier Science Ltd. All rights reserved.

We investigate the non-trivial (associated with the spontaneous symmetry breaking) magnetization plateau at 1/4 of the saturation magnetization of S = 1 spin ladder, especially with reference to recent experimental results on a new organic tetraradical 3,3',5,5'-tetrakis(N-tert, butylaminoxyl)hiphenyl, abbreviated as BIP-TENO. We propose three mechanisms for the formation of the plateau, the Neel mechanism, the dimer mechanism and the spin-Peierls mechanism. We also discuss the effect of four-spin exchange interactions.

We review the numerical diagonalization study on the spin flop in ID higher-S quantum spin chains in the presence of the coupling or single-ion anisotropy. The result suggested that the transition can be second-order for the system which is mapped onto the S = 1/2 XXZ chain. A new result for the S = 1/2 spin ladder is also presented.

Motivated by recent chemical explorations into organic-radical-based higher-spin ladder systems, we study the ground-state properties of a wide class of antiferromagnetic spin-1 ladders. Numerical analysis featuring the level-spectroscopy technique reveals the rich phase diagram, correcting a preceding nonlinear-sigma-model prediction. A variational analysis well interprets the phase competition with particular emphasis on the re-entrant phase boundary on the way from single to coupled chains.

The effect of the four-spin cyclic exchange interaction at each plaquette in the S = 1/2 two-leg spin ladder is investigated at T = 0, focusing especially on the field-induced gap. The strong-rung-coupling approximation suggests that it yields a plateau at half of the saturation moment (m = 1/2) in the magnetization curve, which corresponds to a field-induced spin gap with a spontaneous breaking of the translational symmetry. A precise phase diagram at m = 1/2 is also presented, based on the level spectroscopy analysis of the numerical data obtained by the Lanczos method. The boundary between the gapless and plateau phases is confirmed to be of the Kosterlitz-Thouless (KT) universality class.

The field-induced long-range antiferromagnetic order in the quasi-one-dimensional S = 1 antiferromagnet is investigated with the exact diagonalization of finite chains and the mean field approximation for the interchain interaction. In the presence of the single-ion anisotropy D, we present the phase diagrams in the plane of the magnetic field H and the temperature T for H parallel toD and H perpendicular toD. It is found that the two boundary curves of the ordered phase for H parallel toD and H perpendicular toD have two intersections in the case of D >0 (XY like), while no intersection in the case of D <0 (Ising like). (C) 2001 American Institute of Physics.

Quantum and thermal behaviors of low-dimensional mixed-spin systems are investigated with particular emphasis on the design of molecule-based ferromagnets. One can obtain a molecular ferromagnet by assembling molecular bricks so as to construct a low-dimensional system with a magnetic ground state and then coupling the chains or the layers again in a ferromagnetic fashion. Two of thus-constructed quasi-one-dimensional bimetallic compounds are qualitatively viewed within the spin-wave treatment, one of which successfully grows into a bulk magnet, while the other of which ends in a singlet ground state. Then, concentrating on the ferrimagnetic arrangement on a two-leg ladder which is well indicative of general coupled-chain ferrimagnets. we develop the spin-wave theory and fully reveal its low-energy structure. We inquire further into the ferromagnetic aspect of the ferrimagnetic ladder numerically calculating the sublattice magnetization and the magnetic susceptibility. There exists a moderate coupling strength between the chains in order to obtain the most ferromagnetic ferrimagnet.

We propose a possible mechanism of the charge stripe order due to the next-nearest-neighbor exchange interaction J′ in the two-dimensional t-J model, based on the concept of the phase separation. We also calculate some hole correlation functions of the finite cluster of the model using the numerical diagonalization, to examine the realization of the mechanism. It is also found that the next-nearest-neighbor hopping t′ suppresses the stripe order induced by the present mechanism for t′ &lt 0, while it enhances for t′&gt 0. © 2001 The American Physical Society.

We study the magnetization plateau at 1/4 of the saturation magnetization of the S = 1 antiferromagnetic spin ladder both analytically and numerically, with the aim of explaining recent experimental results on 3,3',5,5'-tetrakis(N-tert-butylaminoxyl) biphenyl. We propose two mechanisms for the plateau formation and clarify the plateau phase diagram on the plane of the coupling constants between spins.

The dynamical local lattice distortion in the CuO2 plane of the cuprate high-temperature superconductors is investigated with the numerical diagonalization study for finite clusters of t-J model. It is found that the in-plane vibrations of the oxygens play an important role in the formation of some static orders like CDW and SDW through the polaronic self localization effect. The 1/8 problem is also discussed based on the present result.

Magnetization processes of ferrimagnetic Heisenberg chains of alternating spins S and s at T = 0 are studied, by the exact diagonalization and size scaling analyses. The results suggest that the spin-(S,s) system has 2s plateaux in the magnetization curve. It is also found that the second and higher plateaux are purely based on a quantum nature, in contrast to the first one that is not distinguished from classical one due to the Ising gap. © 2001 Elsevier Science B.V. All rights reserved.

The pseudogap phenomena observed on cuprate high temperature superconductors are investigated based on the exact diagonalization method on the finite cluster t-J model. The results show the presence of the gap-like behavior in the temperature dependence of various magnetic properties: the NMR relaxation rate, the neutron scattering intensity and the static susceptibility. The calculated spin correlation function indicates that the pseudogap behavior arises associated with the development of the antiferromagnetic spin correlation with decreasing the temperature. The numerical results are presented to clarify the model parameter dependence, that covers the realistic experimental situation. The effect of the next-nearest neighbor hopping t' is also studied.

The direct ESR absorption between the singlet ground and triplet excited states of the spin-gap systems is investigated. Such an absorption, which is forbidden by the conservation of the total spin quantum number in the ordinary Heisenberg-Hamiltonian, is caused by the Dzyaloshinskii-Moriya interaction. We show the selection rule in the presence of the interaction, using the exact diagonalization of the finite cluster of the quasi-one-dimensional bond-alternating spin system. The selection rule is also modified into a suitable form for the investigation of the recent experimental result for CuGeO3. (C) 2001 Elsevier Science B.V. All rights reserved.

The results of magnetic susceptibility and high held magnetization measurements on a single crystal sample of copper zigzag chain compound (N2H5)CuCl3 are reported. Magnetic susceptibility of this compound shows a round maximum around 10K and decreases gradually. High-field magnetization up to 300kOe of this compound shows a monotonic increase with a concave curvature. We compared the experimental results with those of calculation by the temperature-dependent density-matrix-renormalization-group method and obtained fairly good agreement between them when the ratio of the next-nearest-neighbor exchange constant to the nearest neighbor one is 0.2 similar to 0.3. (C) 2001 Elsevier Science B.V. All rights reserved.

Magnetic properties of the low-doped high-T-c cuprates are studied, based on the numerical diagonalization of the finite cluster t-J model. We show the pseudogap behavior observed in the temperature dependence of the NMR relaxation rate, the neutron scattering intensity and the magnetic susceptibility, etc. associated with the presence of the antiferromagnetic spin correlation. The effect of the next-nearest neighbor hopping t' is also presented. (C) 2000 Elsevier Science Ltd. All rights reserved.

The processes of magnetization of ferrimagnetic Heisenberg chains of alternating spins are studied theoretically. Size-scaling analysis with the exact diagonalization of finite systems for (S, s) = (3/2, 1) and (2, 1) indicates a multi-plateau structure in the ground-state magnetization curve for S and s > 1/2. The first plateau in the spontaneous magnetization can be explained classically: as originating from the Ising gap. In contrast, the second and higher ones must be originating from the quantization of the magnetization. It is also found that all of the 2s plateaux, both classical and quantum ones, appear even in the isotropic case with no bond alternation.

The quasi-one-dimensional S = 1 antiferromagnet in magnetic field H is investigated with the exact diagonalization of finite chains and the mean-field approximation for the interchain interaction. In the presence of the single-ion anisotropy D, the full phase diagram in the HT plane is presented for H∥D and H⊥D. The shape of the field-induced long-range ordered phase is revealed to be quite different between the two cases, as observed in the recent experiment of Ni(C5H14N2)2N3(PF 6), (NDMAP). The estimated ratio of the interchain and intrachain couplings of NDMAP (J′/J ∼ 10-3) is consistent with the neutron-scattering measurement.

Ground-state magnetization curves of ferrimagnetic Heisenberg chains of alternating spins S and s are numerically investigated. Calculating several cases of (S,s), we conclude that the spin-(S,s) chain generally exhibits 2s magnetization plateaus even at the most symmetric point. In the double- or more-plateau structure, the initial plateau is generated on a classical basis, whereas the higher ones are based on a quantum mechanism.

The magnetization process of the S = 1/2 antiferromagnetic spin ladder with the next-nearest-neighbor exchange interaction at T = 0 is studied by the exact diagonalization of finite clusters and size scaling analyses. It is found that a magnetization plateau appears at half the saturation value due to the frustration in the presence of sufficiently large rung coupling. The magnetization curves are also presented in several cases. (C) 2000 Elsevier Science B.V. All rights reserved.

The magnetic properties of the antiferromagnetic spin ladder with the next-nearest neighbor interaction, particularly under external field, are investigated by the exact diagonalization of the finite clusters and size scaling techniques. It is found that there exist two phases, the rung-dimer and rung-triplet phases, not only in the nonmagnetic ground state but also magnetized one, where the phase boundary has a small magnetization dependence. Only in the former phase, the magnetization curve is revealed to have a possible plateau at half the saturation moment, with a sufficient frustration. (C) 2000 American Institute of Physics. [S0021- 8979(00)22608-X].

The low-energy structure and the thermodynamic properties of ferrimagnetic Heisenberg chains of alternating spills S and s are investigated by the use of numerical tools as well as the spin-wave theory. The elementary excitations are calculated through an efficient quantum Monte Carlo technique featuring imaginary-time correlation functions and are characterized in terms of interacting spin waves. The thermal behavior is analyzed with particular emphasis on its ferromagnetic and antiferromagnetic dual aspect. The extensive numerical and analytic calculations lead to the classification of the one-dimensional ferrimagnetic behavior according to the constituent; spins: the ferromagnetic (S > 2s), antiferromagnetic (S < 2s), and balanced (S = 2s) ferrimagnetism.

The direct electron spin resonance (ESR) absorption between a singlet ground state and the triplet excited states of spin gap systems is investigated. Such an absorption, which is forbidden by the conservation of the total spin quantum number in Isotropic Hamiltonians, is allowed by the Dzyaloshinskii-Moriya interaction. We show a selection rule in the presence of this interaction, using the exact numerical diagonalization of the finite cluster of the quasi-one-dimensional bond-alternating spin system. The selection rule is also modified into a suitable form in order to interpret recent experimental results on CuGeO3 and NaV2O5.

The magnetization process of the S = 1/2 antiferromagnetic spin ladder at T = 0 is studied by the exact diagonalization of finite clusters and size scaling analyses. It is found that a magnetization plateau appears at half the saturation value (m = 1/2) in the presence of a sufficiently large next-nearest-neighbor exchange interaction to yield the frustration, when the rung coupling is larger than the leg one. The phase diagram at m = 1/2 is given by the analysis based on the conformal invariance. The magnetization curves are also presented in several cases.

We study the magnetization process of the S = 1/2 antiferromagnetic spin ladder in the presence of the second and the third-neighbor couplings which lead to frustration with the typical nearest-neighbor coupling. We use degenerate perturbation theory and level spectroscopy analysis of the numerical diagonalization data of the Hamiltonian for finite systems. We find two kinds of plateaux at half the saturation moment in the magnetization curve. One is mainly due to the second-neighbor couplings and the other to the third-neighbor couplings. The mechanisms of these two plateaux are quite different with each other.

Quantum critical behavior in the magnetization process of alternating-spin chains is studied by the exact diagonalization and the size scaling technique based on the conformal field theory. The plateau at the spontaneous magnetization is revealed to vanish due to the anisotropy at the Kosterlitz-Thouless transition point in the ferromagnetic region. It is also found that the bond alternation creates a novel plateau as an essential quantum effect.

Magnetic properties of the low-doped high-Tc cuprates are investigated based on the numerical diagonalization of the finite cluster t-J model. We show the pseudogap behavior of the temperature dependence of the NMR relaxation rate, neutron scattering intensity and magnetic susceptibility. We also discuss on the origin of the gap-like behavior based on the calculated spin correlation functions. © 2000 Elsevier Science B.V. All rights reserved.

We study the magnetization process of the S = 1/2 antiferromagnetic spin ladder with next-nearest neighbor interaction by the exact diagonalization of finite clusters and by size-scaling technique. It is found that a magnetization plateau appears at half the saturation value due to the field-induced spin gap stabilized by the frustration, when the leg coupling is sufficiently small in comparison with the rung. © 2000 Elsevier Science B.V. All rights reserved.

Ground-state magnetization curves of ferrimagnetic Heisenberg chains of alternating spins S and s are numerically investigated. Calculating several cases of (Formula presented) we conclude that the spin-(Formula presented) chain generally exhibits (Formula presented) magnetization plateaus even at the most symmetric point. In the double- or more-plateau structure, the initial plateau is generated on a classical basis, whereas the higher ones are based on a quantum mechanism. © 2000 The American Physical Society.

We numerically investigate the critical behavior of the spin-(1, 1/2) Heisenberg ferrimagnet with anisotropic exchange coupling in a magnetic field. A quantized magnetization plateau as a function of the field, appearing at a third of the saturated magnetization, is stable over:the whole antiferromagnetic coupling region. The plateau vanishes in the ferromagnetic coupling region via the Kostellitz-Tnouless transition. Comparing the quantum and classical magnetization curves, we elucidate what an essential quantum effects. [S0163-1829(99)00130-7].

We discuss the critical behaviour of the spin-(1, 1/2) Heisenberg ferrimagnetic chain in a magnetic field, whose magnetization curve exhibits a plateau at a third of the full magnetization. A bond alternation stabilizes the massive state, whereas an exchange anisotropy causes the breakdown of the plateau and the onset of a gapless spin-fluid state, where the transition, lying in the XY but ferromagnetic region, is of Kosterlitz-Thouless type. In order to elucidate significant quantum effects, we investigate the model of the classical version as well.

The magnetization process of the one-dimensional antiferromagnetic Heisenberg model with the Ising-like anisotropic exchange interaction is studied by the exact diagonalization technique. It results in the evidence of the first-order spin-flop transition with a finite magnetization jump in the Néel ordered phase for S &gt 1. It implies that the S = 1/2 chain is an exceptional case where the metamagnetic transition becomes second order due to large quantum fluctuations. © 1999 The American Physical Society.

The field induced transition corresponding to the spin flop in the quantum antiferromagnetic chains in the presence of the Ising-like single-ion anisotropy D(&lt 0) is studied by the finite-cluster analysis. It is found that for S&gt 32 a first-order metamagnetic transition occurs even in one dimension for small negative D except for the Haldane phase, while there exist two second-order transitions and an incommensurate phase appears between them for intermediate D, as already found for S=1. We also discuss some experiments of CsNiCl3 related with the present work. © 1999 The American Physical Society.

The magnetization process of the S=1/2 antiferromagnetic spin ladder with the four-spin cyclic exchange interaction at T=0 is studied by the exact diagonalization of finite clusters and size scaling analyses. It is found that a magnetization plateau appears at half the saturation value if the ratio of the four- and two-spin exchange coupling constants J4 is larger than the critical value J4c=0.05±0.04. The phase transition with respect to J4 at J4c is revealed to be the Kosterlitz-Thouless type. © 1999 The American Physical Society.

We numerically investigate the critical behavior of the spin-(1,1/2) Heisenberg ferrimagnet with anisotropic exchange coupling in a magnetic field. A quantized magnetization plateau as a function of the field, appearing at a third of the saturated magnetization, is stable over the whole antiferromagnetic coupling region. The plateau vanishes in the ferromagnetic coupling region via the Kosterlitz-Thouless transition. Comparing the quantum and classical magnetization curves, we elucidate what are essential quantum effects. © 1999 The American Physical Society.

We give a theoretical explanation for the phenomenon of the magnetic susceptibility showing a maximum at low temperature T-m for those compounds that show itinerant-electron metamagnetism. We also derive the linear relation between the metamagnetic transition field H-m and T-m observed experimentally for Y(Co1-xAlx)(2), paying particular attention to the effects of the zero-point quantum spin fluctuations and assuming the existence of almost energetically degenerate paramagnetic thermodynamic states.