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Insights into nature of a magnetization plateau of 3d-4f coordination polymer [Dy2Cu2]n from a spin-1/2 Ising-Heisenberg orthogonal-dimer chain

The ground state and magnetization process of an exactly solved spin-1/2 Ising-Heisenberg orthogonal-dimer chain with two different gyromagnetic factors of the Ising and Heisenberg spins are investigated in detail. It is shown that the investigated quantum spin chain exhibits up to seven possible ground states depending on a mutual interplay of the magnetic field, intra- and inter-dimer coupling constants. More specifically, the frustrated and modulated quantum antiferromagnetic phases are responsible in zero-temperature magnetization curves for a zero magnetization plateau, the intermediate 1/11- and 5/11-plateaus emerge due to the frustrated and modulated quantum ferrimagnetic phases, while the intermediate 9/11- and 10/11-plateaus can be attributed to the quantum and classical ferrimagnetic phases. It is conjectured that the magnetization plateau experimentally observed in a high-field magnetization curve of 3d-4f heterobimetallic coordination polymer [{Dy(hfac)2(CH3OH)}2{Cu(dmg)(Hdmg)}2]n (H2dmg = dimethylglyoxime; Hhfac = 1,1,1,5,5,5-hexafluoropentane-2,4-dione) could be attributed to the classical and quantum ferrimagnetic phases.


Introduction
The frustrated spin- 1 2 Heisenberg orthogonal-dimer (or dimer-plaquette) chain [1][2][3][4] has attracted considerable attention as it represents one-dimensional counterpart of the famous Shastry-Sutherland model [5], which is widely studied by virtue of elucidation a peculiar sequence of fractional plateaus experimentally observed in low-temperature magnetization curves of SrCu 2 (BO 3 ) 2 [6] and rare-earth tetraborides RB 4 [7]. It has been argued by Schulenburg and Richter [8,9] that a zero-temperature magnetization curve of the spin- 1 2 Heisenberg orthogonal-dimer chain displays an infinite series of fractional magnetization plateaus at rational numbers n 2n+2 = 1 4 , 1 3 , . . . , 1 2 , whereas the lowermost and uppermost plateaus from this series are the widest ones. Another interesting feature of the spin- 1 2 Heisenberg orthogonal-dimer chain lies in that it belongs to a prominent class of flat-band models, for which low-temperature thermodynamics can be elaborated by making use of an effective lattice-gas description as extensively discussed by Derzhko and coworkers [10,11].
Regrettably, thermodynamic properties of quantum Heisenberg spin models are in general inaccessible by exact calculations at nonzero temperature. By replacing some of the quantum Heisenberg spins by the classical Ising ones paves the way to an exact solution of the analogous Ising-Heisenberg models by Cu 2 ] n (see the text for a full chemical formula) adapted according to crystallographic data reported in reference [21]. Large cyan balls determine crystallographic positions of Dy 3+ magnetic ions, while small green balls stand for crystallographic positions of Cu 2+ magnetic ions (a color scheme for atom labeling is presented in the legend); (b) The magnetic structure of the corresponding spin- 1 2 Ising-Heisenberg orthogonal-dimer chain, in which Dy 3+ magnetic ions are treated as the Ising spins and Cu 2+ magnetic ions as the Heisenberg spins. The coupling constants J I , J ′ I and J H are assigned to the Ising inter-dimer interaction between Dy 3+ and Cu 2+ magnetic ions (solid lines), the Ising intra-dimer interaction between Dy 3+ magnetic ions (dashed lines) and the Heisenberg intra-dimer interaction between Cu 2+ magnetic ions (dotted lines), respectively. employing exact mapping transformations and the transfer-matrix method [12][13][14]. A few exactly solved versions of the spin- 1 2 Ising-Heisenberg orthogonal-dimer chain [15][16][17][18][19] for instance brought a deeper insight into the magnetization process [15,17,19], magnetocaloric effect [15,17], low-temperature thermodynamics [16][17][18] and bipartite thermal entanglement [16,19] of this frustrated quantum spin chain. Besides, it has been demonstrated that the exact solution for the spin- 1 2 Ising-Heisenberg orthogonaldimer chain may serve as a useful starting point for the many-body perturbation treatment of the fully Heisenberg counterpart model within a more advanced strong-coupling approach [20].
Although we are currently not aware of any experimental realization of the spin-1 2 Heisenberg orthogonal-dimer chain, it surprisingly turns out that 3d-4f heterobimetallic coordination polymer [{Dy(hfac) 2 (CH 3 OH)} 2 {Cu(dmg)(Hdmg)} 2 ] n (H 2 dmg = dimethylglyoxime; Hhfac = 1,1,1,5,5,5hexafluoropentane-2,4-dione) [21], hereafter abbreviated as [Dy 2 Cu 2 ] n , provides an experimental realization of the spin-1 2 Ising-Heisenberg orthogonal-dimer chain. In fact, the polymeric compound [Dy 2 Cu 2 ] n displays a peculiar one-dimensional architecture with regularly alternating dimeric units of Dy 3+ -Dy 3+ and Cu 2+ -Cu 2+ magnetic ions stacked in an orthogonal fashion with respect to each other as illustrated in figure 1(a) [21]. It should be pointed out, moreover, that Dy 3+ magnetic ion represents Kramers ion with the ground-state multiplet 6 H 15/2 , which is subject to a relatively strong crystal-field splitting into eight well-separated Kramers doublets [22,23]. In this regard, the magnetic behaviour of Dy 3+ magnetic ion with the total angular momentum J = 15/2 and the associated g-factor g J = 4/3 can be approximated at low enough temperatures by the classical Ising spin with the effective gyromagnetic factor g Dy = 20 when ignoring admixture of all excited Kramers dublets [22,23]. Getting back to a magnetic structure of the polymeric complex [Dy 2 Cu 2 ] n , which is schematically drawn in figure 1(b), the vertical dimer of Dy 3+ -Dy 3+ magnetic ions may be approximated by a couple of the Ising spins, while the horizontal dimer of Cu 2+ -Cu 2+ magnetic ions may be approximated by a couple of the Heisenberg spins.
The organization of this article is as follows. In section 2 we will describe the studied spin-1 2 Ising-Heisenberg orthogonal-dimer chain and recall basic steps of its exact analytical solution. The ground state and magnetization process of the investigated quantum spin chain will be theoretically studied in section 3. The available experimental data for the high-field magnetization curve of the polymeric compound [Dy 2 Cu 2 ] n will be interpreted by virtue of the spin- 1 2 Ising-Heisenberg orthogonal-dimer chain in section 4. Finally, several conclusions and future outlooks will be mentioned in section 5.

Spin-1 2 Ising-Heisenberg orthogonal-dimer chain
Let us start by introducing the spin-1 2 Ising-Heisenberg orthogonal-dimer chain in a magnetic field through the following Hamiltonian [see figure 1(b) for a schematic illustration]: ,i ) denote standard spin-1 2 operators ascribed to Cu 2+ magnetic ions approximated by the notion of quantum Heisenberg spins andσ z 1(2),i refer to the z-component of the standard spin-1 2 operator ascribed to Dy 3+ magnetic ions approximated by the notion of classical Ising spins. The coupling constants J H and J ′ I determine a strength of the Heisenberg and Ising intra-dimer interactions within the horizontal Cu 2+ -Cu 2+ and vertical Dy 3+ -Dy 3+ dimers, respectively, while the coupling constant J I determines a strength of the Ising inter-dimer interaction between the nearestneighbour Cu 2+ and Dy 3+ magnetic ions. The Zeeman's terms h H = g H µ B B and h I = g I µ B B take into account a magnetostatic energy of magnetic moments relating with the Heisenberg and Ising spins in presence of the external magnetic field B, which differ due to different gyromagnetic factors g H and g I of Cu 2+ and Dy 3+ magnetic ions, respectively.
It is worthwhile to remark that the partition function, Gibbs free energy and magnetization of the spin-1 2 Ising-Heisenberg orthogonal-dimer chain defined through the Hamiltonian (2.1) has been exactly calculated under the periodic boundary conditionsσ z 1(2), N +1 ≡σ z 1(2),1 using the transfer-matrix approach in our preceding paper [19], to which the readers interested in further calculation details are referred to. However, all presented numerical results were restricted just to the particular case h H = h I [19], which corresponds to setting the same Landé g-factors for Cu 2+ and Dy 3+ magnetic ions what is in contrast with the expected (typical) values of the gyromagnetic factors g H ≈ 2 for Cu 2+ magnetic ions and g I ≈ 20 for Dy 3+ magnetic ions. In the present article we will herefore adapt the exact solution for the spin-1 2 Ising-Heisenberg orthogonal-dimer chain reported in reference [19] in order to investigate the effect of different 'local magnetic fields' h H h I arising from the difference of the gyromagnetic factors of Cu 2+ and Dy 3+ magnetic ions g H g I .

Theoretical results
In this section we will examine in detail the ground state and magnetization process of the spin-1 2 Ising-Heisenberg orthogonal-dimer chain by assuming the gyromagnetic factors g H = 2 and g I = 20, which are close to typical values of the Landé g-factors for Cu 2+ and Dy 3+ magnetic ions, respectively. To reduce the number of free parameters a size of the coupling constant J I > 0 corresponding to the antiferromagnetic Ising inter-dimer interaction between Dy 3+ -Cu 2+ magnetic ions will serve as an energy unit when defining a relative strength of the Heisenberg intra-dimer interaction J H /J I within the horizontal dimers, the Ising intra-dimer interaction J ′ I /J I within the vertical dimers and the magnetic field µ B B/J I .

Ground state
The spin- phase |I with an antiparallel spin arrangement of the Ising and Heisenberg dimers characterized through the following eigenvector and corresponding eigenenergy the modulated quantum antiferromagnetic phase |II with alternating character of the Ising dimers and a singlet-like state of the Heisenberg dimers the quantum ferrimagnetic phase |III with fully polarized Ising dimers and a perfect singlet-dimer state of the Heisenberg dimers and finally, the saturated paramagnetic phase |IV with a fully polarized nature of the Ising as well as Heisenberg dimers Note that all eigenvectors are written as a tensor product over states of the vertical Ising dimers (former state vector) and the horizontal Heisenberg dimers (the latter state vector), respectively. At low enough magnetic fields the classical ferrimagnetic phase |I dominates in the parameter region with the ferromagnetic Heisenberg intra-dimer coupling J H < 0, while the modulated quantum antiferromagnetic phase |II and the quantum ferrimagnetic phase |III dominate in the parameter region with the antiferromagnetic Heisenberg intra-dimer coupling J H > 0. Of course, the saturated paramagnetic phase |IV represents the actual ground state at high enough magnetic fields regardless of whether the ferromagnetic or antiferromagnetic Heisenberg intra-dimer coupling is considered.
On the other hand, the ground-state phase diagram of the spin-1 2 Ising-Heisenberg orthogonal-dimer chain with a relatively weak antiferromagnetic Ising intra-dimer coupling additionally involves another two ground states [see figure 2(b) for J ′ I /J I = 0.5], which could be identified as the frustrated quantum antiferromagnetic phase |V with a two-fold degenerate antiferromagnetic state of the Ising dimers and a perfect singlet-dimer state of the Heisenberg dimers 5) and the highly degenerate modulated quantum ferrimagnetic phase |VI with alternating ferroantiferromagnetic character of the Ising dimers and a singlet-like state of the Heisenberg dimers It is quite clear from figure 2(b) that the ground-state phase diagram has not been changed in the parameter space with the ferromagnetic Heisenberg intra-dimer coupling J H < 0, while two novel ground states emerge at low up to moderate magnetic fields in the parameter space with sufficiently strong antiferromagnetic Heisenberg intra-dimer coupling J H > 0. It is noteworthy that the frustrated quantum antiferromagnetic phase |V suppresses the modulated quantum antiferromagnetic phase |II upon increasing of a relative strength of the antiferromagnetic Ising intra-dimer coupling J ′ I /J I until this latter ground state completely vanishes from the ground-state phase diagram as evidenced by figure 2(c) for J ′ I /J I = 2.0. Last but not least, the sufficiently strong antiferromagnetic Ising intra-dimer coupling J ′ I /J I > 2 may additionally cause uprise of the novel ground state also in the parameter space with the ferromagnetic Heisenberg intra-dimer coupling J H < 0 and small enough magnetic fields, which could be identified as the frustrated ferrimagnetic phase |VII with a two-fold degenerate antiferromagnetic state of the Ising dimers and fully polarized Heisenberg dimers given by

Zero-and low-temperature magnetization curves
The gapped ground states Now, a few comments are in order as far as a thermal stability of the individual magnetization plateaus is concerned. It is quite obvious from figure 3 that some intermediate plateaus are quite robust with respect to thermal fluctuations and they can be clearly discerned in the respective isothermal magnetization curves at low (k B T/J I = 0.05) or even moderate (k B T/J I = 0.15) temperatures. This is typically the case for the zero magnetization plateau and the intermediate 9/11-and 10/11-plateaus emergent at higher values of the magnetization. On the other hand, the intermediate 1/11-and 5/11-plateaus emergent at smaller values of the magnetization are typically subject to considerable smoothing upon increasing of temperature and they cannot be clearly discerned in the respective isothermal magnetization curves even at relatively low temperature (k B T/J I = 0.05). This striking finding can be related to a substantial difference of the energy gap of the individual ground states ascribed to the relevant magnetization plateaus. As a matter of fact, the gapped ground states as for instance the classical and quantum ferrimagnetic phases |I and |III possess due to their unique nature a high energy gap in an excitation spectrum. Contrary to this, the highly degenerate ground states such as the frustrated ferrimagnetic phase |VII and the modulated quantum ferrimagnetic phase |VI possess only a tiny energy gap and hence, they are subject to a substantial thermal smoothing owing to their macroscopic degeneracies.

Magnetization curve of the polymeric compound [Dy 2 Cu 2 ] n
In this part we will compare available experimental data for the magnetization curve of the 3d-4f heterobimetallic coordination polymer [Dy 2 Cu 2 ] n measured in pulsed magnetic fields up to 10 T at temperature T = 0.5 K [21] with a relevant theoretical prediction based on the spin-1 2 Ising-Heisenberg orthogonal-dimer chain. It is worth noticing that all five model parameters of the investigated quantum spin chain defined through the Hamiltonian (2.1) were supposed as free fitting parameters in order to get the best theoretical fit of the experimental magnetization data (see figure 4 for a comparison). The best theoretical fit was accordingly obtained for the following set of the model parameters: J I /k B = 8.02 K, J ′ I /k B = 17.35 K, J H /k B = 1.73 K, g H = 2.28 and g I = 18.54. The reported value of the gyromagnetic factor g H = 2.28 is quite typical for Cu 2+ magnetic ions, while the other reported value g I = 18.54 is only by a few percent (cca. 7%) lower than the value g Dy = 20 theoretically expected for the effective Ising-spin description of Dy 3+ magnetic ions with the total angular momentum J = 15/2 and the respective g-factor g J = 4/3 [22,23]. It should be pointed out, moreover, that the values of the Ising inter-and intra-dimer coupling constants J I and J ′ I should be also rescaled by the factors of 15 and 225 in order to get true values of the coupling constants between Dy 3+ -Cu 2+ and Dy 3+ -Dy 3+ magnetic ions when passing away from the effective Ising-spin description of Dy 3+ magnetic ions. The actual values of the three considered coupling constants consequently read: J Dy−Cu /k B = 0.53 K, J Dy−Dy /k B = 0.08 K and J Cu−Cu /k B = 1.73 K, whereas the predicted values of the coupling constants J Dy−Cu and J Dy−Dy fall into a reasonable range for the coupling constants between 3d-4f and 4f-4f magnetic ions, respectively [22,23]. Note furthermore that the former exchange constant is comparable with the mean value of the exchange coupling J Dy−Cu /k B = 0.48 K, which can be calculated from the coupling constants J A /k B = 0.895 K and J B /k B = 0.061 K assigned previously to two different exchange pathways between Dy 3+ -Cu 2+ magnetic ions within the simplified tetranuclear model [21]. It is worthwhile to remark that the reported value for the coupling constant J Cu−Cu /k B = 1.73 K is relatively small with respect to the shortest distance between Cu 2+ -Cu 2+ magnetic ions, but this small value can be attributed to a rather inefficient exchange pathway involving a double-oxygen bridge between magnetic (3d x 2 −y 2 ) and nonmagnetic (3d  four field-driven phase transitions |V − |VI − |I − |III − |IV . A low-field part of the magnetization curve should accordingly display two narrow plateaus at zero and approximately 5/11 of the saturation magnetization (note that the gyromagnetic factors slightly deviate from the ideal values g H = 2.0 and g I = 20.0), which are however completely smeared out by thermal fluctuations at temperatures as low as T 0.5 K due to tiny energy gaps of the phases |V and |VI . As a matter of fact, the zero magnetization plateau corresponding to the frustrated quantum antiferromagnetic phase |V is restricted to the magnetic fields smaller than 0.17 T, while the 5/11-plateau related to the modulated quantum ferrimagnetic phase |VI is limited to the magnetic-field range 0.17 − 0.28 T.
On the other hand, the high-field part of the magnetization curve should exhibit two wider intermediate plateaus roughly at 9/11-and 10/11 of the saturation magnetization. The 9/11-plateau pertinent to the classical ferrimagnetic phase |I is being stable within the magnetic-field range 0.28 − 4.12 T, while the 10/11-plateau ascribed to the quantum ferrimagnetic phase |III is being stable within the magneticfield range 4.12 − 6.37 T. In this regard, it is therefore quite puzzling that the field-driven transition between the classical and quantum ferrimagnetic phases cannot be evidently seen from the magnetization data recorded for the polymeric compound [Dy 2 Cu 2 ] n at relatively low temperature T = 0.5 K as both ground states |I and |III with a substantial energy gap should be quite resistant with respect to a thermal smoothing. It is plausible to conjecture two possible reasons for the discrepancy between the experimental magnetization data recorded at temperature T = 0.5 K and the respective theoretical fit: either the sample of the polymeric compound [Dy 2 Cu 2 ] n was not kept during the magnetization process under a perfect isothermal condition since small heating (e.g. due to the magnetocaloric effect) could resolve this discrepancy as evidenced by a theoretical curve calculated for slightly higher temperature T = 1.0 K, or the discrepancy is of intrinsic origin and it comes from two different exchange pathways between Dy 3+ -Cu 2+ magnetic ions neglected within the investigated quantum spin chain.

Conclusion
In the present article we have investigated in detail diversity of ground-state phase diagrams and magnetization curves of the spin- 1 2 Ising-Heisenberg orthogonal-dimer chain by assuming two different gyromagnetic factors of the Ising and Heisenberg spins. The investigated quantum spin chain including two different Landé g-factors was inspired by the polymeric coordination compound [Dy 2 Cu 2 ] n , whose magnetic structure shows a peculiar one-dimensional architecture with regularly alternating dimeric units of Dy 3+ -Dy 3+ and Cu 2+ -Cu 2+ magnetic ions stacked in an orthogonal fashion [21]. The vertical dimer of Dy 3+ -Dy 3+ magnetic ions was approximated by a couple of the Ising spins, while the horizontal dimer of Cu 2+ -Cu 2+ magnetic ions was approximated by a couple of the Heisenberg spins. It has been found that the spin-1 2 Ising-Heisenberg orthogonal-dimer chain exhibits a rich variety of the classical and quantum ground states, which involve besides the fully saturated paramagnetic phase emergent at sufficiently high magnetic fields additional six ground states: the frustrated and modulated quantum antiferromagnetic phase, the frustrated and modulated quantum ferrimagnetic phase, as well as, the quantum and classical ferrimagnetic phase. These remarkable ground states are responsible for presence of magnetization plateaus in zero-and low-temperature magnetization curves, which are manifested at 0, 1/11, 5/11, 9/11 and/or 10/11 of the saturation magnetization. A stability of the intermediate magnetization plateaus with respect to thermal fluctuations was investigated in detail. It has been verified that the rather narrow 1/11-and 5/11-plateaus ascribed to the frustrated and modulated quantum ferrimagnetic phases with a high macroscopic degeneracy and a tiny energy gap are easily destroyed upon a small increase of temperature, while the relatively wide 9/11-and 10/11-plateaus ascribed to the nondegenerate classical and quantum ferrimagnetic phases with a robust energy gap are quite resistant with respect to thermal fluctuations and may be discernible also at relatively low up to moderate temperatures.
Subsequently, we have successfully applied the spin-1 2 Ising-Heisenberg orthogonal-dimer chain with two different gyromagnetic factors of the Ising and Heisenberg spins for a theoretical modeling of the high-field magnetization data recorded previously for the polymeric coordination compound [Dy 2 Cu 2 ] n at sufficiently low temperature T = 0.5 K [21]. The respective low-temperature magnetization curve evidently displays intermediate magnetization plateau(s), which starts at a relatively high value of the magnetization being approximately 78% of the saturated value. The best theoretical fit of the available experimental data based on the spin-1 2 Ising-Heisenberg orthogonal-dimer chain suggests that the magnetization plateau(s) observed experimentally could be ascribed to the classical and quantum ferrimagnetic phases given by equations (3.1) and (3.3) inherent to the intermediate 9/11-and 10/11plateau. It should be mentioned, however, that the magnetization data measured for the polymeric complex [Dy 2 Cu 2 ] n at low enough temperature T = 0.5 K do not provide any experimental evidence for the relevant field-induced phase transition between the classical and quantum ferrimagnetic phase [21]. This discrepancy could be resolved either by a small deviation from a perfect isothermal condition during the experimental measurement caused by heating of the sample (e.g. due to the magnetocaloric effect) or it may relate to the oversimplified nature of the introduced spin-1 2 Ising-Heisenberg orthogonaldimer chain neglecting two structurally inequivalent exchange pathways between Dy 3+ -Cu 2+ magnetic ions. In our future work we plan to examine the magnetization process of the spin-1 2 Ising-Heisenberg orthogonal-dimer chain taking into consideration two different exchange constants between Dy 3+ -Cu 2+ magnetic ions in order to clarify this issue.