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On the absence of actual plateaus in zero-temperature magnetization curves of quantum spin clusters and chains

We examine the general features of the non-commutativity of the magnetization operator and Hamiltonian for the small quantum spin clusters. The source of this non-commutativity can be a difference in the Land\'{e} g-factors for different spins in the cluster, XY-anisotropy in the exchange interaction and the presence of the Dzyaloshinskii-Moriya term in the direction different for the direction of the magnetic field. As a result, a zero-temperature magnetization curve for the small spin clusters mimics that for the macroscopic systems with the band(s) of magnetic excitations, i.e. for the given eigenstate of the spin cluster the corresponding magnetic moment can be an explicit function of the external magnetic field yielding the non-constant (non-plateau) form of the magnetization curve within the given eigenstate. Also, the XY-anisotropy makes the saturated magnetization (the eigenstate when all spins in cluster are aligned along the magnetic field) inaccessible for finite magnetic field magnitude (asymptotical saturation). We demonstrate all these features on three examples: spin-1/2 dimer, mixed spin-(1/2,1) dimer, spin-1/2 ring trimer. We consider the simplest Ising-Heisenberg chain, the XYZ-Ising diamond chain with four different g-factors as well. In the chain model the magnetization curve has more complicated and non trivial structure with than that for clusters.


I. INTRODUCTION
Magnetization curves of low-dimensional quantum antiferromagnets are topical issue of current research interest, because they often involve intriguing features such as magnetization plateaus, jumps, ramps and/or kinks. The spin-1/2 quantum Heisenberg chain, the spin-1/2 quantum Ising chain in a transverse field, and the spin-1/2 quantum XX chain in a transverse field are a few paradigmatic examples of exactly solved quantum spin chains for which zero-temperature magnetization varies smoothly with rising magnetic field until the saturation magnetization is reached. [1][2][3] Contrary to this, the integervalue quantum Heisenberg chains (and also many other low-dimensional quantum antiferromagnets) contain in a zero-temperature magnetization process remarkable magnetization plateau(s) at rational value(s) of the saturation magnetization. 4,5 The intermediate plateaus of Heisenberg spin chains reflect quantum states of matter with exotic topological order such as the Haldane phase, 6,7 whereas their presence is restricted by quantization condition known as Oshikawa-Yamanaka-Affleck rule. 8,9 On the other hand, it could be generally expected that the antiferromagnetic Heisenberg spin clusters should always exhibit leastwise one intermediate plateau before the magnetization jumps to its saturation value. [10][11][12][13] This naive expectation follows from the energy spectrum of the quantum Heisenberg spin clusters, which is com-posed of a few discrete energy levels that cannot naturally form a continuous energy band needed for a smooth variation of the magnetization at zero temperature. At first sight, this argumentation is consistent with the existence of at least one plateau and magnetization jump, which bears a close relation to level crossing caused by the external magnetic field. From this perspective, the quite natural question arises whether or not intermediate magnetization plateau(s) can be partially or completely lifted from zero-temperature magnetization curves of the Heisenberg spin clusters.
Another spin systems which should be noted in the context of the small quantum spin clusters are the Ising-Heisenberg chains. They are the one-dimensional spin systems where the small quantum spin clusters are assembled to the chain by alternating with the Ising spins in such a way, that the Hamiltonian for the whole system is a sum of mutually commuting block Hamiltonians. These systems have much in common with the "classical" chains of the Ising spins, as they can be solved by the same technique and the eigenstates are just the direct product of the eigenstates of the single block, though, for the more complicated structure the doubling of unit cell is possible. The magnetization curves, thus, for the Ising-Heisenberg spin systems share almost all features with the magnetization curves of the small spin clusters, but can contain much more intermediate magnetization plateaus. Various variants of the Ising-Heisenberg chains have been examined: diamond-chain, [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] saw-tooth chain, 29,30 orthogonal-dimer chain, [31][32][33] tetrahedral chain, [34][35][36][37][38] and some special examples relevant to real magnetic materials. [39][40][41][42] In the present work, we will rigorously examine a magnetization process of a few quantum Heisenberg spin clusters and Ising-Heisenberg diamond chain, which will not display strict magnetization plateaus on assumption that some constituent spins have different Landé g-factors and may be a XY-anisotropy of the exchange interaction. Also, the Dzyaloshinskii-Moriya (DM) term in a direction different from that of the magnetic field can lead to the same effect. All those features of the spin Hamiltonian make the magnetization non-conserved, i.e. non-commutating with the Hamiltonian. This specific requirement naturally leads to a nonlinear dependence of the energy levels on a magnetic field, which consequently causes a smooth change of the magnetization with the magnetic field within one and the same eigenstate. Although the smooth change of magnetization due to a difference in Landé g-factors or/and XY-anisotropy and non-collinear DM-term may be quite reminiscent of that of quantum spin chains with continuous energy bands, it is of course of completely different mechanism with much simpler origin.
The single-chain magnet, is a remarkable example of both Ising-Heisenberg one-dimensional spin system and a spin model with different Landé g-factors, leading to nonplateau form of the region of the magnetization curve corresponding to the same eigenstate. However, as the exact analysis shows 42 the effect is just hardly visible in the magnetization curve plot in virtue of the very small difference in Landé g-factors of the magnetic ions, though, the exact expression for the magnetization has explicit dependence on the magnetic field. Almost the same effect but even quantitatively less pronounced have been observed in the approximate model of the onedimensional magnet, the F-F-AF-AF spin chain compound Cu(3-Chloropyridine) 2 (N 3 ) 2 . 39 The organization of this paper is as follows. In the next Section, we will clarify a few general statements closely related to an absence of actual plateaus in zerotemperature magnetization curves of quantum spin clusters and chains. These arguments of general validity will be subsequently illustrated on a few specific examples of the spin-1/2 quantum Heisenberg dimer, the mixed spin-(1/2,1) Heisenberg dimer, the spin-1/2 Heisenberg trimer and the spin-1/2 Ising-Heisenberg diamond chain in the following four Sections. The summary of the most important findings along with the implications for experimental systems will be presented in the concluding part.

II. GENERAL STATEMENTS
Let us first start with a few very general statements elucidating the issue of the non-constant magnetization within one physical state or the explicit magnetic field dependence of the magnetization corresponding to a certain eigenstate of the small spin clusters. Obviously, the aforementioned phenomenon arises when the magnetization is not a good quantum number Here, H stands for the Hamiltonian of a spin cluster. One can distinguish two cases, when z-projection of the total spin S z tot does not commute with the Hamiltonian and the magnetization operator is proportional to it or when the z-projection of the total spin S z tot is a good quantum number, but the magnetization operator is not proportional to it and does not commute with the Hamiltonian Of course, another possibility is to have the magnetization which is non proportional to S z tot and the zprojection of the total spin S z tot non-conserved. The spin Hamiltonians, which do not commute with S z tot , usually contain XY-anisotropy or/and DM vector with a nonzero X or Y part. The magnetization is non-proportional to the total spin S z tot when the spins possess different Landé g-factors.

III. SPIN-1/2 HEISENBERG DIMER
In this section we consider the spin-1/2 Heisenberg dimer as the simplest system of two interacting quantum spins described by the most general Hamiltonian Here, S α 1,2 , (α = x, y, z) are the spatial components of the spin-1/2 operators for two spins in the dimer. We assume the fully anisotropic XYZ Heisenberg coupling with two anisotropy constants γ, ∆ and two different but isotropic Landé g-factors. The spatial direction of the magnetic field B and the DM-vector D are arbitrary so far. Without loss of generality, one may however choose a direction of the magnetic field along the z-axis and the DM vector to lie in xz-plane Let us calculate the commutators of the Hamiltonian (5) with the z-projections of the operators corresponding to the total spin and magnetization where g ± = g 1 ± g 2 . As one can see, the XY-anisotropy, γ, and the DM-vector x-projection, D x , make the S z tot and M z non conserved, but even if we set them to zero, the magnetization may still be a non-conserved quantity because of the difference in Landé g-factors. Thus, the spin-1/2 Heisenberg dimer may exhibit the non-constant magnetization within one ground state if at least one of the parameters, g 2 − g 1 , γ or D x (D y ) is non-zero. Let us put D x = 0 as it makes the analytic calculations quite cumbersome (the eigenvalue problem leads to a quartic equation), and start with the exact diagonalization of the Hamiltonian for the anisotropic spin-1/2 Heisenberg dimer with different Landé g-factors. The eigenvalues are The corresponding eigenvectors are A ± =ρ ± e iφ , φ = arctan D z J , Under the conditions g 1 = g 2 (and D z = 0) the first two eigenstates become conventional singlet and S z tot = 0 component of the triplet respectively. However, there is no continuous transition to the S z tot = 1 and S z tot = −1 components of the triplet in |Ψ 3,4 at γ → 0. In order to obtain | ↑↑ and | ↓↓ one has to put γ = 0 in the Hamiltonian before diagonalization. Let us calculate the magnetization eigenvalues for all those eigenstates: At g 1 = g 2 this expression becomes 0. However, at g 1 = g 2 we have explicit dependence of the eigenvalue, corresponding to the certain eigenstate on the magnetic field. This leads to a non constant magnetization for the given eigenstate. Thus, we have here S z tot = 0 and M z = 0.
For the other two eigenstates we have At γ = 0 the expression transform to which corresponds to | ↑↑ and | ↓↓ eigenstates. The Eq. (11) has another important feature. The transverse quantum fluctuations enhanced by the XY-anisotropy γ reduce the magnetization in z-direction in such a way that it never reaches its saturated values ± 1 2 (g 1 + g 2 ) at any non-zero γ and finite magnetic field B. It is also important that even at the equal g-factors, g 2 = g 1 = g the magnetization expectation values for the eigenstates |Ψ 3,4 exhibit explicit magnetic field dependence and do not reach their saturated values at non-zero γ. Another case of interest is the g 2 = −g 1 = g, when the magnetization expectation value for the |Ψ 1,2 is non-zero and exhibits explicit dependence on the magnetic filed and the eigenstates |Ψ 3,4 demonstrate zero magnetization. Moreover, the corresponding expectation values become singular at γ = 0, because, as was mentioned above, there is no continuous limit γ → 0 in terms of eigenvalues and eigenvectors. The magnetic susceptibility for the aforementioned states can be obtained in a straightforward way by taking a derivative of the Eqs. (10) and (11) with respect to B: The deviation form the horizontal line for the zerotemperature magnetization curve of the dimer under consideration, thus, is governed by three factors. Difference of Landé g-factors, g 2 − g 1 is responsible for the nonplateau behavior for the initial part of the magnetization curve, B ≤ B c , the larger is the absolute value of the difference the more pronounced the deviation is. At the same time, the overall Landé g-factor, g 1 + g 2 and the XY-anisotropy γ make another part of the magnetization curve, which in the limit γ = 0 correspond to the saturation, | ↑↑ , non flat. The critical field B c is found from the level crossing. As for the zero temperature only |Ψ 2 and |Ψ 4 are realized we can found the corresponding value of the magnetic field from the equation, ε 2 = ε 4 , which leads to with The typical picture of the level crossing curve one can see in Fig. 1. However, the B = 0 ground state is also affected by the value of the XY-anisotropy γ. The ground state becomes |Ψ 4 for a sufficiently large γ above certain critical value γ c . The critical value is given by the equation The non-linear behavior with respect to the magnetic field is the main reason for the non-plateau magnetization. As the DM-term in z-direction does not bring any qualitatively new physics, we can hereafter put D z = 0. The expression for the critical field (14) does not lead to a proper γ = 0 limit. The case of isotropic Heisenberg interaction must be considered separately. In the case of isotropic Heisenberg interaction the difference is only in the saturated states presented here, and which transforms to |Ψ 3,4 at non-zero γ, while the |Ψ 1,2 eigenstates remain the same. The value of critical field in this case is Thus, the jump to the saturated magnetization takes place for the γ = 0 at this value of the magnetic field. The magnitude of the jump depends on the difference of the Landé g-factors and is given by The corresponding plots of the zero-temperature magnetization one can find in the Figs. 2 and 3. In the Fig. 2 the evolution of the T = 0 ground state for different values of γ are presented for J = 1, D z = 1, ∆ = 2, g 1 = 2 and g 2 = 6. The critical value of γ at which the B = 0 ground state of the spin-1/2 spin dimer changes from |Ψ 2 to |Ψ 4 for these values of J, D z and ∆ is γ c = 2 + √ 2 ≃ 3.41. Therefore, for γ = 0 and γ = 2 one can see magnetization curves with two eigenstates separated by the jump. The non-plateau behavior of the magnetization for |Ψ 2 at B < B c is well visible. Also, the non-plateau character of the magnetization curve, corresponding to |Ψ 4 is obvious for γ = 2, while for γ = 0 we see ideal plateau at M = 1 2 (g 1 + g 2 ) = 4. This non-plateau behavior and inaccessibility of the saturation are more pronounced for γ = 4 and γ = 6 when the system for all values of the magnetic field (B > 0) is in the |Ψ 4 eigenstate and its T = 0 magnetization curve demonstrates the form very similar to the one of a system with a band of magnetic excitations or/and to the high-temperature magnetization curve given by the Brillouin function. The effect of the difference between the Landé g-factor is summarized in Fig. 3. To demonstrate the evolution of the ground state |Ψ 2 under the change of the difference of Landé g-factors, we have chosen γ = 0 and J = 1, D z = 0, ∆ = 1 and plotted the normalized magnetization M/M sat , as saturation magnetization, M sat = 1 2 (g 1 +g 2 ) is different for each curve. For the g 1 = g 2 there are just two ideal plateaus at M = 0 (singlet state) and M = 1. The magnetization jumps from 0 to 1 at B c = J 1+∆ 2g (here g 1 = g 2 = g). However, the essential changes appear when the difference between g-factors is growing. For g 2 − g 1 non equal to zero the part of the magnetization curve corresponding to the S z 1 + S z 2 = 0 deviates from the horizontal line and become almost linear (for small g − ) and then more and more rapidly growing with the shift of the transition point between |Ψ 2 and |Ψ 4 in the lower B region.
Of course, the effects of the DM-terms in molecular magnets and other low-dimensional many-body spin systems have been intensively studied in various contexts during last decade. [43][44][45][46][47] In Ref.44 the isolated spin-dimer with DM-terms has been considered with general mutual orientation of the DM-vector and magnetic field, where D = (0, D sin θ, D cos θ). Though, as was shown above, in this case the eigenvalue problem leads to the solution of quartic equation, the authors found an approximate ground state in the limit D/J ≪ 1 and below the critical field B = J/g. They explicitly found out the magnetization of the ground state which turned out to be linear in B This is the approximate form of the non-linear behavior of the magnetization we have obtained exactly above in the case of D = (0, 0, D z ). Despite of all these results, the issue of the non-conserving magnetization and its consequences have not been systematically investigated so far.
It is also straightforward to construct the thermodynamics of the simple dimer. The partition function is calculated directly from the spectrum: The magnetization is found in a standard way, as The plots of the finite-temperature magnetization for the S=1/2 spin dimer are presented in Fig. 4. It is worth mentioning that thermal fluctuations eliminate from magnetization curves all typical structures (such as plateaus or quasi-plateaus) quite similarly as the large XY-anisotropy does for zero-temperature magnetization curves.

IV. MIXED-SPIN HEISENBERG DIMER
Another interesting example of a simple quantum spin system, which may possibly show a striking dependence of the total magnetization on a magnetic field, is The normalized zero-temperature magnetization curves, M/Msat of the S = 1/2 dimer with two different g-factors in case of isotropic exchange interaction, γ = 0 and ∆ = 1 . Here, for the sake of simplicity, we put J = 1, Dz = 0, g1 = 2 and present the curves for the different values of g2. From the bottom to top g2=2 (red); 4 (green); 6 (blue); 8 (magenta); 10 (brown) and 20 (black). Msat = 1 2 (g1 + g2). the mixed spin-(1/2,1) Heisenberg dimer defined by the Hamiltonian Here, S α 1 and µ α 2 (α = x, y, z) represent spatial components of the spin-1/2 and spin-1 operators, respectively, the exchange constant J denotes the XXZ Heisenberg coupling between the spin-1/2 and spin-1 magnetic ions, ∆ is an exchange anisotropy in this interaction, D is an uniaxial single-ion anisotropy acting on a spin-1 magnetic ion, g 1 and g 2 are Landé g-factors of the spin-1/2 and spin-1 magnetic ions in an external magnetic field B. As the effect of the XY-anisotropy was described in details in the previous Section, here to put it simple, we assume γ = 0. A straightforward diagonalization of the Hamiltonian (23) gives a full spectrum of eigenstates, which can be characterized by the following set of eigenvalues and the corresponding eigenvectors where the respective probability amplitudes are given by It should be mentioned that the mixed spin-(1/2,1) Heisenberg dimer exhibits a strict intermediate plateau at one-third of the saturation magnetization regardless of uniaxial single-ion anisotropy on assumption that the Landé g-factors of both constituent magnetic ions are equal g 1 = g 2 . If the Landé g-factors are different g 1 = g 2 , then, the mixed spin-(1/2,1) Heisenberg dimer displays more intriguing zero-temperature magnetization curves basically affected by a relative strength of the uniaxial single-ion anisotropy. To illustrate the case, the total normalized magnetization of the mixed spin-(1/2,1) dimer is plotted in Fig. 5 against the magnetic field for the isotropic Heisenberg coupling ∆ = 1, several values of the uniaxial single-ion anisotropy and two particular sets of Landé g-factors. A smooth variation of the total magnetization observed below a saturation field relates to a gradual change of probability amplitudes of two entangled microstates | − 1/2, 1 and |1/2, 0 within the eigenstate |Ψ 3 . In a low-field region with continuously varying magnetization, mean values of two constituent spins and the total magnetization can be therefore calculated with the help of the corresponding lowest-energy eigenvector |Ψ 3 given by Eq. (25) At g − = 0 we have here M = g 2 (g 1 = g 2 = g). It can be seen from Fig. 5 that the respective field variations of the total magnetization (normalized by the saturation magnetization M sat = 1 2 (g 1 + 2g 2 )) depend basically on whether the Landé g-factor of the spin-1 magnetic ion is greater or smaller than the g-factor of the spin-1/2 magnetic ion. The total magnetization is gradually suppressed by an increase in the single-ion anisotropy in the former case g 1 < g 2 (see Fig. 5a), while the total magnetization is conversely enhanced by an increase in the single-ion anisotropy in the latter case g 1 > g 2 (see 5b). In general, the total magnetization displays a considerable dependence on a magnetic field for small enough single-ion anisotropies D/J ≈ 0, while one recovers a quasi-plateau dependence with only a subtle variation of the total magnetization in two limiting cases D/J → ±∞ at which the following asymptotic values are reached However, the most surprising zero-temperature dependence of the total magnetization can be found when the g-factor of the spin-1/2 magnetic ion is much greater than the g-factor of the spin-1 magnetic ion (g 1 ≫ g 2 ) and the uniaxial single-ion anisotropy is of easy-axis type D < 0. It turns out that the eigenvector |Ψ 5 characterized by a quantum entanglement of two microstates |1/2, −1 and |−1/2, 0 may eventually become the lowest-energy eigenstate with a positive value of the total magnetization in spite of negative value of the total spin S z tot = −1/2. It is quite evident that a strong enough easy-axis singleion anisotropy suppresses the occurrence probability of the microstate | − 1/2, 0 , whereas the other microstate |1/2, −1 may lead to a positive magnetization due to much greater the Landé g-factor of the spin-1/2 magnetic ion g 1 ≫ g 2 than that of the spin-1 magnetic ion. Mean values of two constitutent spins and the total magnetization follow from the corresponding lowest-energy eigenvector |Ψ 5 given by Eq. (25) In the analogy with the |Ψ 3 case, here we get M = − g 2 at g 1 = g 2 = g. The zero-temperature magnetization curves displayed in Fig. 6 afford a convincing proof that the total magnetization may vary continuously in a lowfield region, then it may show an abrupt jump to an intermediate-field region with another continuously varying magnetization terminating just at the saturation field (see the magnetization curves for D/J = −0.4 and −0.5).
In accordance with the previous argumentation, the total magnetization follows the formula (28) in the low-field region attributable to the eigenstate |Ψ 5 , while it varies according to Eq. (29) in the intermediate-field region attributable to the eigenstate |Ψ 3 . The magnetization part corresponding to the eigenstate |Ψ 3 gradually diminishes as the easy-axis single-ion anisotropy strengthens (i.e. it becomes more negative) and hence, the total magnetization shows below a saturation field only a single region with continuously varying magnetization due to the striking lowest-energy eigenstate |Ψ 3 with a negative total spin but a positive total magnetization (see the magnetization curves for D/J = −0.6 and −1.0). It is straightforward to calculate the susceptibility for the eigenstates |Ψ 3 , and |Ψ 5 : which become zero only when g 1 = g 2 or D/J → ∞ or ∆ → ∞. It has been demonstrated that a smooth variation of the total magnetization at zero temperature within one and the same eigenstate requires a difference between the Landé g-factors. From this perspective, our theoretical predictions could be more easily ex- perimentally tested for the mixed spin-(1/2,1) Heisenberg dimer, which represents plausible model for heterobimetallic dinuclear complexes naturally having two unequal Landé g-factors due to two different constituting magnetic ions. While the quasi-plateau phenomenon should still remain a rather subtle effect in heterodinuclear complexes composed of Cu 2+ (spin-1/2) and Ni 2+ (spin-1) magnetic ions due to a relatively small difference between the g-factors not exceeding a few percent, 61,62 it should become much more pronounced in heterodinuclear complexes composed of Co 3+ (spin-1/2) and Ni 2+ (spin-1) magnetic ions having much greater difference between g-factors (typically g Co ≈ 5.9 and g Ni ≈ 2.3). 52,53,63

V. SPIN-1/2 HEISENBERG TRIMER
The next by complicity spin system with the different g-factors is the triangle with uniform coupling and with only two g-factors given by the Hamiltonian The Hamiltonian can be diagonalized in a straightforward way. The eigenvalues are: where The eigenvectors are where the number in the lower right angle of the symbol | correspond to the certain spin in the triangle, and |S , |T ± and |T 0 are spin-singlet and components of the spin-triplet: And the coefficients are For our purposes the eigenvectors from |Ψ 5 to |Ψ 8 are of special interest, as they demonstrate the monotonous explicit dependence of the magnetization on the magnetic filed under constant value of the projection of the total spin, which is ± 1 2 in our case. The expectation value of the magnetic moment for the eigenstate with the lowest energy among the others (|Ψ 8 for positive B) is It is easy to see that for the case of the uniform g-factors, g 1 = g 2 = g the magnetic moment expectation value become a constant equal to g/2. Though, for the eigenvalues of the magnetization operator the limit g − = 0 gives the correct result, this is not the case for the eigenvectors of the Hamiltonian. Thus, one cannot obtain the standard basis for the spin trimer by putting g 1 = g 2 in the Eqs, (34). The susceptibility for the continuous magnetization (37) is given by The susceptibility becomes zero at g 1 = g 2 . The partition function for the spin-trimer under consideration can be obtained in a straightforward way: The finite-temperature magnetization reads The plots of the normalized zero-temperature magnetization are presented in Fig. 7. Here the development of the non-plateau part of the magnetization curve with the increase in the difference g 2 − g 1 is clearly visible. For comparison the ordinary curve for g 1 = g 2 is also presented with a plateau at M/M sat = 1/3, which corresponds to the ground state with S 2 tot = 3/4, S z tot = 1/2: which transforms into the |Ψ 8 at g 2 = g 1 . Let us mention also, that the B = 0 ground state at for g 1 = g 2 is four-fold degenerate and the magnetic field lifts this degeneration just partly, because the 1/3 plateau state is still two-fold degenerate. The deviation from the horizontal line becomes more pronounced with the growing difference between g-factors of the spins. Thus, the zerotemperature magnetization curve for the simple system with finite discrete spectrum mimics the magnetic behavior of magnets with the band of magnetic excitations. The value of critical field at which the level crossing be-tween |Ψ 8 and the fully polarized state |Ψ 1 takes place is In the limit g 1 = g 2 = g the value of critical field is J 1+2∆ 2g . The interplay between thermal fluctuations and the non-plateau behavior can be seen in the Fig. 8 where one can see a gradual smearing out of the magnetization curve with the rise of the temperature.

VI. ISING-HEISENBERG (ISING-XYZ) DIAMOND CHAIN WITH DIFFERENT LANDÉ G-FACTORS
To illustrate the features of having the spin cluster with non-conserving magnetization as a constituent of the Ising-Heisenberg one-dimensional systems let us now consider the simplest Ising-Heisenberg spin-chain with the XYZ-dimers and different g-factors, the diamond chain. 14-28 However, we are not going to describe the whole problem in all details, this can be a topic of the forthcoming and separate investigation. We just want to illustrate how rich the structure of the magnetization curve can be, if we include the spin cluster with nonconserved magnetization into the more involved structures. The interest toward the diamond chain is large not only because of the simplicity of the system, especially in case of Ising-Heisenberg one-dimensional systems, but Here we also consider the g-factors of the quantum spins Sj,1 and Sj,2 to be g1 and g2 respectively. also as the diamond chain is believed to be the real magnetic structure of the mineral azurite. [48][49][50][51] The lattice in depicted in Fig. 9, where the quantum spin-dimer are the vertical bonds(solid lines) while the dashed lines correspond to Ising couplings. The Hamiltonian for the whole chain is the sum over the block Hamiltonians: where the g-factors of the Ising intermediate spins alternating with the spin-dimer are also taken alternating g j = g 3 , j is odd g 4 , j is even.

A. Exact solution
One can apply the standard technique of the generalized classical transfer matrix to calculate the free energy of the model under consideration exactly. 16,18,22,[29][30][31]35,36,42 However, as here we deal with the alternation of two kind of blocks one has to compose a two block transfer matrix just by multiplying transfer matrices for odd and even blocks. In the other words, the partition function of the model can be factorized in the following form (the cyclic boundary conditions are assumed): where with the aid of the block Hamiltonian eigenvalues from Eq. (8), one can obtain four quantities V σj ,σj+1 , σ j , σ j+1 = ±1/2, as the entries of the following matrix: U 0 e −β B(g 3 +g 4 ) where Thus, the partition function can be rewritten in the form where, Then, for the free energy per unit cell in the thermodynamic limit, N → ∞, we have, where the λ is the largest eigenvalue of the matrix T, which is expressed by the entries of the matrix T in the following form: Now, we will analyze the non-plateau magnetization for Ising-XYZ diamond chain with different g-factors.
Using the free energy, one gets the magnetization As the lattice has six spins in the translational invariant unit cell, two σ spins and two vertical dimers, the total saturation magnetization per one block (note that N is the number of block which is supposed to be even, while the number of the unit cell with six spins is N/2) is One can see the plots of the zero-temperature magnetization of the Ising-XYZ diamond chain in Fig. (10). The zero-temperature curves can be obtained as a sufficiently low-temperature limit of the exact expression (51). One can see series of quasi-plateau and magnetization jumps which can undergo a certain variation under the changing of the XY-anisotropy γ (upper panel) or axial anisotropy ∆ (lower panel).

B. Ground states
We are not going to present the comprehensive analysis of all possible ground states and all types of magnetization curves for the XYZ-Ising diamond-chain with different g-factors, but let us just illustrate the ground state structure of the T = 0 magnetization curve from Fig. (10a) To be specific let us consider the magnetization curve for J = 1, K = 1, g 1 = 8, g 2 = 2, g 3 = 2, g 4 = 4, γ = 0.5 and ∆ = 1.2. The comparison of the different combinations of the spin-dimer eigenstates (Eq. (9) with D z = 0) with the orientation od the intermediate Ising spins gives rise to series of possible eigenstates for the chain. However, only few of them are realized in the case we considered here. First of all, the B = 0 ground state is macroscopically degenerate for our choice of parameters. In this ground state all vertical quantum dimers are in |Ψ 2 state form Eq. (9) and all intermediate Ising spins can freely point either up or down. Thus, we have here 2 N configuration with the same energy. Arbitrary but non-zero magnetic field lifts this degeneracy. The system passes through the following ground state with the increasing the magnetic field which correspond to the course of the magnetization curve we are analyzing here: where the eigenstates with the corresponding magnetization and energies are here, |Ψ 2,4 j stand for the corresponding eigenvectors of the spin-dimer (Eq. (9) with D z = 0) at j-th block and the |Ψ ± 4 j is structurally the same |Ψ 4 j eigenstate but taking into account the influence of the interaction with the neighboring Ising spins σ j and σ j+1 , which leads only to a modification of the coefficient B − : the arrows indicate the orientation of the corresponding Ising spins. Here several comments are in order. First of all, one can see the phenomenon of reentrant phase transition when the system with increasing the magnitude of the magnetic field enters the same ground state, |GS1 , twice. It also clearly can be seen from the ground state phase diagram which is presented in Fig. (11) . The return to the ground state |GS1 is possible due to non-linear magnetic field dependence of the corresponding magnetization and the energies of all ground states. One can also see from the Eq. (54) that despite the visible ideal horizontal character of some part of the magnetization curve, they are not the magnetization plateaus, but just the very slowly growing part of the curve. Thus, the magnetization always has an explicit dependence on the magnetic field, unless γ = 0 or/and g 1 = g 2 . The quasisaturated state, |QS , at finite values of γ has the magnetization asymptotically converging to M sat . However, this value is inaccessible for finite values of magnetic field. Although most of quasi-plateaus appear below the saturation (last critical) field, there also exists an alternative mechanism for quasi-plateau formation. Namely, the last plateau emerging above the last critical field (which is in fact not a true saturation field) may change to the quasiplateau due to the XY anisotropy and consequently, the magnetization varies continuously above the last criti-cal field and it never reaches full saturation except for asymptotically infinite magnetic field.

VII. CONCLUSION
In the present work, we have investigated in detail an absence of actual plateaus in zero-temperature magnetization curves of quantum spin clusters and chains. It has been convincingly evidenced that strict plateaus may disappear from a magnetization process on assumption that constituent spins of quantum spin clusters have different Landé g-factors. More specifically, we have demonstrated this intriguing feature on a few paradigmatic examples of quantum spin clusters such as the spin-1/2 Heisenberg dimer, the mixed spin-(1/2,1) Heisenberg dimer and the spin-1/2 Heisenberg trimer, whereas the same phenomenon has been also found in the spin-1/2 Ising-Heisenberg diamond chain. From this perspective, the absence of actual magnetization plateaus due to the difference in Landé g-factors can be regarded as a general feature of low-dimensional quantum antiferromagnets, because it emerges whenever the total magnetization does not represent a conserved quantity with well defined quantum spin numbers (the total magnetization need not be proportional to the total spin). Accordingly, a smooth variation of total magnetization can be attributed to a nonlinear dependence of a few (or all) discrete energy levels on a magnetic field.
A few remarks are in order here, which might be useful for possible experimental testing of this interesting phenomenon. Although the magnetization curves of quantum spin clusters without strict plateaus may mimic to a certain extent the magnetization curves of quantum spin chains with continuous energy bands, it is obvious that the absence of magnetization plateaus does not in turn mean a gapless excitation spectrum. On the contrary, small magnetic spin clusters should always have an energy gap, which can be easily experimentally tested by various resonance techniques.
It is also worth noting that the deviation of magnetization from a strict plateau is proportional to a difference between the Landé g-factors, which makes an experimental verification of this phenomenon more difficult. As a matter of fact, the most of transition-metal ions as for instance Cu 2+ , Ni 2+ , Mn 2+ , Cr 3+ or high-spin Fe 3+ with zero or totally quenched angular momentum can be described by the notion of more or less isotropic quantum Heisenberg spins and hence, these magnetic ions usually have g-factors quite close to the free electron value g ≈ 2. [52][53][54] Under these circumstances, it is customary to combine the local Landé g-factors of individual magnetic ions into a single molecular g-factor as long as the isotropic Heisenberg exchange significantly prevails over the zero-field splitting, asymmetric and/or antisymmetric exchange. 55 Many experimental studies focused on a magnetism of such oligonuclear complexes therefore simply ignore different Landé g-factors of individual magnetic ions as the isotropic exchange is by far the most dominant coupling. However, a few transitionmetal ions with unquenched angular momentum may have much higher Landé g-factors due to a relatively strong spin-orbit coupling like for example the low-spin Fe 3+ ion with typical value of g ≈ 2.8 or Co 2+ ion with g ≈ 6.0. [52][53][54] Another possibility how to increase the difference of the Landé g-factors in oligonuclear complexes is to combine the almost isotropic transition-metal ion with highly anisotropic rare-earth ions, which may even possess much greater Landé g-factors (e.g. Dy 3+ typically has g ≈ 20) though this extraordinary large g-value usually correlates with a rather strong anisotropy in the exchange interaction. 52,56,57 Under the extreme situation, the XY-part of exchange coupling might be even of opposite sign than the Z-part (ferromagnetic versus antiferromagnetic) as it has been recently reported for the heterodinuclear Cr 3+ -Yb 3+ complex. 56 Last but not least, let us briefly comment on experimental implications for the quantum spin clusters studied in the present work. The spin-1/2 Heisenberg dimer has previously proved its usefulness as the plausible model of many homodinuclear Cu 2+ -Cu 2+ complexes. [52][53][54]58 However, the difference between the local Landé g-factors in the homodinuclear coordination compounds may only stem from a different coordination sphere of individual magnetic ions, whereas this difference does not exceed in most cases a few percent that would be insufficient for an experimental testing. On the contrary, the considerable difference in the local g-factors could be expected in heterobimetallic coordination compounds, which are composed of the nearly isotropic magnetic ion (e.g. Cu 2+ , Ni 2+ , high-spin Fe 3+ , etc.) and the highly anisotropic magnetic ion (e.g. Co 2+ , low-spin Fe 3+ , etc.). Hence, the heterodinuclear Co 2+ -Cu 2+ and Fe 3+ -Cu 2+ complexes could be regarded as sought experimental realizations of the generalized spin-1/2 Heisenberg dimer, which may display a substantial deviation of the magnetization from a strict plateau as the g-factors of individual magnetic ions may even possess opposite signs due to a spin-orbit coupling (e.g. g Fe ≈ −1.7, g Cu ≈ 2.1 was reported in Ref. [59], and negative g-factors of Co 2+ and Cu 2+ magnetic ions were investigated in Ref. [60]). The similar conjecture can be formulated for experimental representatives of the mixed spin-(1/2,1) Heisenberg dimer. As usual, the magnetic anisotropy in heterodinuclear Cu 2+ -Ni 2+ complexes does not cause a significant deviation of the magnetization from a strict plateau, 61,62 but a rather large deviation could be expected instead in heterodinuclear Co 2+ -Ni 2+ coordination compounds with typical values of the g-factors g Co ≈ 5.9 and g Ni ≈ 2.3. 63 It can be also anticipated that the homotrinuclear Cu 2+ -Cu 2+ -Cu 2+ complexes 64-66 as experimental representatives of the spin-1/2 Heisenberg trimer should not have a significant deviation of the magnetization from a strict plateau unlike the heterotrinuclear Cu 2+ -Co 2+ -Cu 2+ complexes.